Author:
LeAnn Barr, Shelbyville HS
With Special Thanks To:
Sheri Whiteside, Jacksonville HS
Teresa Ward, Jacksonville MS
The University of Texas at Austin
Charles A. Dana Center
P.O. Box M
Austin, TX 78713
512-471-6190
http://www.utdanacenter.org
for gracious use of:
TEXTEAMS – Algebra 2000 and Beyond
Textbook - p. 4, 5, 8, 9
Workbook – p. 3 - 6, 11 - 14, 17 -19, 97, 119 – 122,
Teacher Resources Activity 4-3A
Algebra I – Assessments
Workbook – p. 115-116
i
Contents
Chapter 1: Patterns
1-1 Solving Equations ………………………………………………. page 2
a. Solving one step equations using Algebra Tiles
b. Relationship Statements
c. Labeling the values of the x- and y-axis.
1-2 Patterns …………………………………………………………… page 4
a. Identifying the variables as Independent or Dependent
b. Plotting points in quadrant 1
c. Finding missing terms in patterns.
d. Finding a reasonable domain and range.
e. Finding the rate of change, fixed number and function rule.
f. Applying function notation to patterns.
1-3 Patterns with Word Problems ……………………………….….page 6
1-4 Patterns with a rate of change greater than 1 ……………… page 8
Solving two-step equations.
Chapter 2: Functions
© Copyright 2006 – LeAnn Barr
2-1 Fun Functions Activity pages ................................................page 12
a. Identifying functions from graphs, coordinates, and mappings
b. Identifying domain and range.
2-2 Function Lesson – “Moving Men” …………………………….page 14
a. Solving equations that involve fractions.
b. Identifying variables as Independent or Dependent.
c. Finding rates of change, fixed number and function rule.
d. Using function notation
e. Plotting Points, identifying quadrants and axis.
f. Using a Table
g. Finding missing values
h. Finding domain and range.
i. Solving two-step equations
2-3 Function Lesson - “TAXI” ………………………………………page 16
2-4 Function Lesson – More Domain and Range ……………….page 18
Algebra I - TEXAS Style
ii
2-5 Function Lesson “Airplane Problems” ………………………page 20
2-6 Function Lesson – “Music Book” ……………………………. page 22
a. Domain and Range
b. Rate of Change, Y-intercept and Function Rules
c. Function Notation
d. Systems of Equations (both already y= )
e. Solving equations with variables on both sides.
f. Finding a break even point and profit. (activities)
2-7 Function Lesson – More equations with variables on both sides
.................................page 24
2-8 Function Lesson – Using Coordinates to find function rules
.................................page 26
Chapter 3: Graphing
3-1 Scatter Plots and Correlation Practice …………………………page 30
a. Plotting scatter plots on the calculator
b. Finding the line of best fit.
c. Identifying correlation from graphs and words.
© Copyright 2006 – LeAnn Barr
3-2 Direct Variation ……………………………………………………..page 32
a. Vocabulary for Direct Variation
b. Constant of Variation and Function Rules
3-3 Graphing Linear Equations – Slope …………………………….page 34
a. Finding Slope and Graphing using slope.
b. Writing equations of lines.
c. Graphing and Comparing to the parent function y=x.
3-4 Graphing Linear Equations – More Slope ……………………..page 36
Graphing a line using slope and a point
3-5 Graphing Linear Equations – y=mx+b …………………………page 38
a. Graphing equations of lines
b. Identifying slope and y-intercept.
Algebra I - TEXAS Style
iii
3-6 Graphing Linear Equations – Domain and Range …………page 40
a. Using domain and range.
b. Vertical shifting of Lines.
3-7 Parallel and Perpendicular Lines ……………………………. page 42
3-8 Distance Between Two Points and Midpoint ……………….page 44
3-9 Standard Form ……………………………………………………page 46
TAKS Practice …………………………………………………………page 48
Chapter 4: Inequalities
4-1 Inequalities with One – Variable …………………………… ..page 50
Solving and Graphing One-Variable Inequalities
4-2 Inequalities with Two – Variables ……………………………page 52
Solving and Graphing Two-Variable Inequalities
4-3 Inequalities Application Problems …………………………..page 54
TAKS Practice ………………………………………………….……..page 56
© Copyright 2006 – LeAnn Barr
Chapter 5: Systems
5-1
Systems and Perimeter ……………………………………..page 58
a. Solving Perimeter Problems
b. Drawing and labeling a diagram
5-2 Systems Finding Solutions …………………………..……...page 60
a. Solving Systems using a Table
b. Solving Systems by Elimination
5-3 Systems and Graphing …………………………….……….. page 62
Solving for y and graphing on the calculator
TAKS Practice ……………………………………………………... page 64
Algebra I - TEXAS Style
iv
Chapter 6: Exponents
6-1 Exponents …………………………………………………..… page 66
Discovering Rules for Exponents by Expanding.
6-2 Exponents and Division ……………………………….…… page 68
a. Division and exponents
b. Scientific Notation
c. Multiplying and Dividing with Scientific Notation
6-3 Negative Exponents ………………………………………… page 70
TAKS Practice ………………………………………………...……page 72
Chapter 7: Polynomials
7-1 Polynomials – Simplifying ……………………………..….page 74
a. Adding and Subtracting Polynomials
b. With and Without Algebra Tiles
7-2
Polynomials using the Distributive Property
……….…….page 76
a. Distributive Property
b. With and Without Algebra Tiles
© Copyright 2006 – LeAnn Barr
7-3 Multiplying Polynomials ………………………………….page 78
With and without Algebra Tiles
TAKS Practice ……………………………………………………..page 80
Algebra I - TEXAS Style
v
Chapter 8: Solving Quadratic Equations
8-1 Exploring Factoring …………………..……………………page 82
a. Finding a common factor, roots, and y-intercepts.
b. Using Algebra Tiles and Graphing.
8-2 Factoring
8-3 Factoring
x 2 ± bx + c ……..…………………………...page 84
x 2 ± bx − c ……………………………..……page 86
8-4 Factoring Squares and Difference of Squares
8-5 Factoring
……..page 88
ax 2 ± bx ± c ………….……………………..page 90
8-6 Solving Quadratic Equations …………..…………………page 92
a. Quadratic Formula
b. Verifying by Graphing
© Copyright 2006 – LeAnn Barr
Index………………………………………………………………page 94
Algebra I - TEXAS Style
vi
Patterns
1-1 Solving Equations ……………………………………………. page 2
a. Solving one step equations using Algebra Tiles
b. Relationship Statements
c. Labeling the values of the x- and y-axis.
d. TEKS a3, a5, b1A, b4A, c3B
New Numbered TEKS a3, a5, A.1A, A.4A, A.7B
e. Workbook Set 1-1
1-2 Patterns ………………………………………………………… page 4
a. Identifying the variables as Independent or Dependent
b. Plotting points in quadrant 1
c. Finding missing terms in patterns.
d. Finding a reasonable domain and range.
e. Finding the rate of change, fixed number and function rule.
f. Applying function notation to patterns.
g. TEKS same as above plus a1, a2, a4, b1B, b1C, b1D, B1E, b2B, b2D,
b3A, b3B, b4A, c1B, c1C, c2A, c2B, c3B
New Numbered TEKS same as above plus a1, a2, a4, A.1B, A.1C, A.1D, A.1E,
A.2B, A.2D, A.3A, A.3B, A.4A, A.5B, A.6A, A.6B, A.7B
h. Workbook Activities 1,2,3 and 4 and Set 1-2
© Copyright 2006 – LeAnn Barr
1-3 Patterns with Word Problems ……………………………….page 6
a. TEKS same as above
b. Workbook Set 1-3
1-4 Patterns with a rate of change greater than 1 …………… page 8
a. Solving two-step equations.
b. TEKS same as above
c. Workbook Activities 5 through 14 and Sets 1-4B and 1-4D
Algebra I - TEXAS Style
1
Patterns and One – Step Equations
Lesson 1-1
Ex. 1 Solve the equation x + 3 = 6 using Algebra Tiles.
Alvin Alligator
What do you have to add to
both sides, so x is by itself?
I’m going to add -3 to both
sides.
x
Legend
1 -x -1
Simplify both sides, to
find the value of x.
Now solve it
Algebraically
Same step
Note:
-3+3=0
X +3 = 6
-3 -3
x=3
Ex. 2 Solve x – 3 = 5 using Algebra tiles and showing algebraic steps.
© Copyright 2006 – LeAnn Barr
Now solve it
Algebraically
X-3=5
+3 +3
x=8
Ex. 3 Solve x + 1 = -3 using Algebra tiles and showing algebraic steps.
Now solve it
Algebraically
X +1 = -3
-1 -1
x = -4
Try these: 1. x + 2 = 4
2. x – 2 = 4
3. x + 2 = -4
Algebra I - TEXAS Style
2
4. x – 2 = -4
Patterns and One – Step Equations
Relationship Statements:
Lesson 1-1
If I complete my homework, then I will
make good grades.
Rewrite the above statement using the following key words:
Depends: “Making good grades depends on doing my homework.”
Is a Function of: “Making good grades is a function of doing my homework.”
Determines: “Doing my homework determines if I make good grades.”
Vocabulary:
Y-Axis
Dependent variable
Range
y-intercept
Output
List the corresponding names.
© Copyright 2006 – LeAnn Barr
Origin
X-Axis
Independent variable
Domain
x-intercept
solutions
roots
Input
TAKS Preparation:
A rectangle has an area of 100 square feet. If the length and width are
both tripled, what is the new area?
A.
B.
C.
D.
33 square feet
100 square feet
300 square feet
900 square feet
Create a rectangle that has an area of 100 sq feet,
then triple both sides.
20
5
new width = 5(3) = 15
new length = 20(3) = 60
New Area = 15(60) = 900
It is 3(3) or 9 times as big.
Algebra I - TEXAS Style
3
Patterns – One Step Patterns
Example 1.
Lesson 1-2
Process
Cubes in
All
1
1+4
5
2
2+4
6
3
3+4
7
4
4+4
8
Term #
(cubes in
the tower)
1
a. Label the axis of the graph and plot the
points.
1
© Copyright 2006 – LeAnn Barr
b. What is the independent variable? Cubes
in the tower (It is the title at the top of the
left hand column)
5
5+4
9
10
20
?
?
n
10+4
20+4
14
24
32
50
n+4
What is the dependent variable? Cubes
in all (It is the title at the top of the right
hand column)
c. What kind of correlation does this have?
Since the dots on the graph go up from
left to right, it has a positive correlation.
d. What is a reasonable domain? Domain represents the independent variable or x. So this
reasonable domain is from 1 to any number you pick as reasonable, like 20. Because it might
fall after 20 cubes in the tower. (Any reasonable number will do, as long as you can justify it)
What is a reasonable range? Range represents the dependent variable
or y. These values depend on what you chose for your domain. See the
table at the right. A reasonable range is 5 to 24 cubes in all.
Domain
Range
1
1+4
5
20
20+4
24
e. What is the rate of change? The total number of cubes is increasing by one for each new cube
1
in the tower. The rate of change is = 1 .
1
What is the fixed number? 4 is the number that is added to the number of cubes in the tower
each time. 4 is the fixed number.
What is the function rule? F(n)=1n+4 or n+4
Pattern Activities Adapted from TEXTEAMS – Algebra 2000 and Beyond,
Used with permission, The University of Texas at Austin, Charles A. Dana Center
Algebra I - TEXAS Style
4
Patterns – One Step Patterns
f.
Find each value:
Lesson 1-2
f(4) = 4+4=8
f(5) = 5+4 =9
f(10) = 10+4 =14
f(20) = 20+4 = 24
f(n) = 32
f(n) = 32
n + 4 = 32
−4 −4
n = 28
Try this one:
Pattern Activity 1
1. Copy and complete the table:
Term #
Process
(cubes
in the
tower)
1
Cubes in
All
3
n + 4 = 50
−4 −4
n = 46
2. Draw and label a graph.
3. What is the independent
variable?
4. What is the dependent
variable?
2
5. What kind of correlation
does this have?
3
6. What is a reasonable
domain?
7. What is a reasonable
range?
4
© Copyright 2006 – LeAnn Barr
8. What is the rate of change?
5
9. What is the fixed number?
Draw this one.
10. What is the function rule?
10
11. Find f(2)
20
12. Find f(4)
?
38
13. Find f(10)
?
103
14. Find f(20)
n
15. Find n if f(n) = 38
16. Find n if f(n) = 103
Pattern Activities Adapted from TEXTEAMS – Algebra 2000 and Beyond,
Used with permission, The University of Texas at Austin, Charles A. Dana Center
Algebra I - TEXAS Style
5
Patterns
Lesson 1-3
Warmup
1) Sue has 3 red, 2 yellow, 4blue, and 3 green ribbons.
a. Find the probability of getting a blue. P(blue).
b. Find P(blue, yellow) without replacement
2) The amount of money I make, depends on how much I work.”
a. Copy the sentence, then underline the independent part and circle the
dependent part.
b. Rewrite using “determines”
Example 1: Jorge is saving money for a new CD stereo player. He has $25 now and
will be saving $1 a week from his allowance. The CD stereo player he wants cost
$59. Answer each:
a. Complete the table:
© Copyright 2006 – LeAnn Barr
# of weeks
Process
Column
0+25
1+25
2+25
3+25
4+25
10+25
15+25
b. Draw and label a graph.
Money
Saved
25
26
27
28
29
35
40
50
59
0
1
2
3
4
10
15
c. What is the independent variable? # of weeks
?
?
d. What is the dependent variable? Money Saved
n
n+25
e. Write a relationship statement. The money Jose has saved depends on the number of weeks
that he has been saving.
f. What kind of correlation does this have? Positive
g. What is the domain and range? The domain is from 0 to 34 weeks because he will have enough
money to buy the CD stereo player in 34 weeks. The range is $25 to $59.
1
h. What is the rate of change? = 1 , it increases by $1 each time.
1
i. What is the fixed number? $25
j. What is the function rule? n+25
k. Find each value:
F(10) = 10+25 = 35
F(15) = 15 + 25 = 40
F(n) = 50
F(n) = 59
n + 25 = 50
-25 -25
n = 25
n + 25 = 59
-25 -25
n = 34
Patterns
Lesson 1-3
Example 2: Jason’s Mom made chocolate chip cookies, oatmeal cookies and sugar cookies. 40%
of the cookies are chocolate chip, and 20% are oatmeal cookies. There are 48 sugar cookies.
a. What percent of the cookies are sugar cookies? 100%-40%-20%=40%
0.40 x
48
=
b. How many cookies did Jason’s Mom make in all? 0.40 0.40 , 120 cookies
x = 120
c. How many chocolate chip cookies did she make? (0.40)(120) = 48
d. How many oatmeal cookies did she make? (0.20)(120) = 24
3
1
1
red beads,
silver beads, blue
10
10
5
beads and there are 12 gold beads. Answer each question:
Example 3: Gabrielle bought a bead set. It contained
a. What is the fractional part for gold beads? Gold = whole – all the other parts
3 1 1
1− − −
10 10 5
10 3 1
2
− − −
10 10 10 10
4 2
=
10 5
© Copyright 2006 – LeAnn Barr
b. What is the total number of beads?
2 12
= , x = 30
5 x
c. How many red beads are in the package?
3
r
,r=9
=
10 30
d. How many silver beads are in the package?
1
s
,s=3
=
10 30
e. How many blue beads are in the package?
1 b
,b=6
=
5 30
Try these; solve these equations, show all your work:
1. 0.50x = 6
2. x + 1.30 = 10
3. 1.5x = 12
4.
2 x
=
3 12
Patterns – Two Step Equations
Example 1:
1. Complete the table.
Lesson 1-4
2. Label and graph the data.
3. What is the independent variable? Tower height
4. What is the dependent variable? # of cubes in all
5. What kind of correlation does this have? positive
6. What is a reasonable domain? Tower height of 1
to 25 cubes (I chose 25 this time because my group
only has 25 cubes and that is all we could build.)
7. What is a reasonable range? Total number of 7 to
55 cubes.
© Copyright 2006 – LeAnn Barr
8. What is the rate of change?
2
= 2 , it is increasing by 2 cubes in all each time.
1
9. What is the fixed number? 5, each time 5 is needed to make the function rule work in the table.
10. What is the function rule? f(x) = 2x + 5
11. Find each value:
a. f(4) = 13
b. f(5) = 15
c. f(10) = 25
See the table for the work on questions a – d.
e. f(n) = 39, n = ?
2x+5=39
-5 -5
2x=34
2 2
x = 17
d. f(50) = 105
f. f(n) = 71, n = ?
2x+5=71
-5 -5
2x=66
2 2
x = 33
Used with permission, The University of Texas at Austin, Charles A. Dana Center
TEXTEAMS: Algebra 2000 and Beyond.
Algebra I - TEXAS Style
8
Patterns – Two Step Equations
Lesson 1-4
Example 2: Solve this equation using Algebra Tiles and show the Algebraic Steps:
First subtract 1(the
number without an x)
from both sides of the
equation.
Then divide what’s left
by the number of x’s
2x + 1 = 5
−1 −1
2x 4
=
2
2
x=2
Try this equation: Draw the Algebra Tiles and Show the Algebraic Steps. 3x – 1 = 5
Try this pattern activity:
Pattern Activity 5
1. Copy and complete the table
Term #
Picture
(# of cubes)
1
2
Faces
to paint
5
9
2. Draw and label a graph of this pattern.
3. What is the independent variable?
4. Dependent variable?
5. What kind of correlation does this have?
© Copyright 2006 – LeAnn Barr
6. What is the domain?
7. What is the range?
3
8. What is the rate of change?
4
5
10
50
?
?
n
9. What is the fixed number?
Draw this
one
10. What is the function rule?
11. Find each value:
a. f(4)=
37
105
d. f(50)=
b. f(5)=
c. f(10)=
e. f(n)=37, what is n?
f. f(n)=105, what is n?
Used with Permission - The University of Texas at Austin, Charles A. Dana Center
TEXTEAMS - Algebra 2000 and Beyond
Algebra I - TEXAS Style
9
Functions
2-1 Fun Functions Activity pages ................................................page 12
a. Identifying functions from graphs, coordinates, and mappings
b. Identifying domain and range.
c. TEKS a2, a3, b2B
New Numbered TEKS a2, a3, A.2B
d. Workbook Activity 2-1 “Fun Functions”
2-2 Function Lesson – “Moving Men” …………………………….page 14
a. Solving equations that involve fractions.
b. Identifying variables as Independent or Dependent.
c. Finding rates of change, fixed number and function rule.
d. Using function notation
e. Plotting Points, identifying quadrants and axis.
f. Using a Table
g. Finding missing values
h. Finding domain and range.
i. Solving two-step equations
j. TEKS a1-6, b1A, b1B, b1C, b1D, b1E, b2B, b2D, b3A, b3B, b4A, c1B, c1C,
c2A, c2B, c3A, c3B, c3C
New Numbered TEKS a1-6, A.1A, A.1B, A.1C, A.1D, A.1E, A.2B, A.2D, A.3A,
A.3B, A.4A, A.5B, A.5C, A.6A, A.6B, A.7A, A.7B, A.7C
© Copyright 2006 – LeAnn Barr
k. Workbook set 2-2
2-3 Function Lesson - “TAXI” ………………………………………page 16
a. TEKS – same as above
b. Workbook Set 2-3
2-4 Function Lesson – More Domain and Range ……………….page 18
a. TEKS – same as above
b. Workbook Set 2-4
2-5 Function Lesson “Airplane Problems” ………………………page 20
a. TEKS – same as above
b. Workbook Set 2-5A and 2-5B
Algebra I - TEXAS Style
10
2-6 Function Lesson – “Music Book” ………………………….. page 22
a. Domain and Range
b. Rate of Change, Y-intercept and Function Rules
c. Function Notation
d. Systems of Equations (both already y= )
e. Solving equations with variables on both sides.
f. Finding a break even point and profit. (activities)
g. TEKS – same as above plus c2F, c4A, c4B, c4C
New Numbered TEKS same as above plus A.6F, A.8A, A.8B, A.8C
h. Workbook Activity Set 2-6A and 2-6B “T-shirts”
2-7 Function Lesson – More equations with variables on both sides
.................................page 24
a. TEKS – same as above
b. Workbook Set 2-7A and 2-7B
2-8 Function Lesson – Using Coordinates to find function rules
.................................page 26
© Copyright 2006 – LeAnn Barr
a. Finding function rules from coordinates of points.
b. TEKS – same as above
c. Workbook Set 2-8A “Plumbing Problems” and 2-8B “Billboards”
Algebra I - TEXAS Style
11
Fun Functions
Lesson 2-1
Domain represents the x-values in a set of numbers, graph or mapping.
Range represents the y-values in a set of numbers, graph or mapping.
Functions:
1
These are functions.
Can you tell why?
2
3
6. {(0,3), (2,7), (5,3)}
© Copyright 2006 – LeAnn Barr
7. {(9,2), (8,2), (7,2)}
Examples 1, 2, and 3: Nothing in
the domain repeats. You can tell
by using a vertical line. Lay your
pencil straight up and down on
the graph. How many times does
it cross? It only crosses once.
1
2
Example 4: The domain is {8,1,4} and nothing in the domain repeats.
Example 5: The domain is {8,4,3} and nothing in the domain repeats.
Example 6: The domain is {0,2,5} and nothing in the domain repeats.
Example 7: The domain is {9,8,7) and nothing in the domain repeats.
Algebra I - TEXAS Style
12
3
Fun Functions
Lesson 2-1
Non-Functions:
These are not functions.
Can you tell why?
9
8
11. {(0,3), (2,7), (0,5)}
Example 8: The domain repeats. You can tell by using a
vertical line. Lay your pencil straight up and down on the
graph. How many times does it cross? It crosses three times
so it is not a function.
8
Example 9: The domain repeats. A vertical line crosses this line everywhere, so
it is not a function.
Example 10: The domain is {12,6,6}. The 6 is used more than once, so it is not a
function.
© Copyright 2006 – LeAnn Barr
Example 11: The domain is {0,2,0}. 0 is used more than once, so it is not a
function.
Definition of a function: A relationship between values where the domain does
not repeat.
Try These:
State whether each is a function:
1
2
5. {(2,3)(9,10),(-4,2),(9,3)}
3
6. {(1,3),(4,2),(5,0),(-1,5)}
Algebra I - TEXAS Style
13
Function Activity 2-2 (Moving Men)
Name:
Warmup
1) Sue bought a quarter sheet cake for $15. How much would the entire cake cost?
2) Which of the following function rules fits the pattern below?
A. y = 5x
B. y = 5x-1
C. y = 5x + 1
D. y = 6x
More Solving Equations:
1.
If half a gallon of ice cream cost $3,
how much is a whole gallon?
It Looks like:
1
x=3
2
Try these:
2.
1
x=7
3
1
x = −10
5
3.
© Copyright 2006 – LeAnn Barr
4. Joanne bought two-fifths of a package of gum for 10 cents. How much would the
entire package cost?
Looks like:
Try these: 5.
3
x=9
4
6.
2
x = 10
5
2
x = 10
3
7. The scale factor for two similar triangles is
2
. If the perimeter of the smaller
3
triangle is 12 inches, what is the perimeter of the larger triangle?
Algebra I - TEXAS Style
14
Function Activity 2-2 (Moving Men)
Name:
8. The Moving Men Moving Company charges a flat fee of $100 plus
$50 per hour.
a. Complete the table:
b. Sketch a graph. Be sure to label both axis.
c. What is the rate of change? How does this rate
of change relate to the problem?
$50 per hour
d. What is the fixed number? How does the fixed number
relate to the graph? $100 flat fee, it is the y-intercept
e. What is the function rule? M(x) = 50x + 100
f.
Graph this function on the calculator and identify a good
viewing window for this problem. Sketch:
X min= 0
X max= 10
Y min= 0
Y max= 700
Your window could be different but should match the information given, and be similar to this
one.
© Copyright 2006 – LeAnn Barr
g. What is the independent variable? What is the domain? How are they related?
Hours. The domain is 0 hours and up, {x>0}. Domain represents the allowed values for
hours.
h. What is the dependent variable? What is the range? How are they related?
Cost. The range is $100 and up, {y>100}. The range represents the allowed values for
cost.
i.
How much would it cost if it took 7 hours to move? 50(7)+100 = $450
j.
How much would it cost if it took 12 hours to move?
Try these:
k. How much would it cost if it took 20 hours to move?
l.
How many hours did it take to move if the cost was $950?
m. How many hours did it take to move if the cost was $1250?
Algebra I - TEXAS Style
15
50x + 100 = 950
-100 -100
50x = 850
50 50
x = 17
This activity is in
your workbook.
Functions Activity “Taxi”
Lesson 2-3
The Yellow Taxi Cab Company charges a flat fee of $2.50 plus
$0.75 per mile.
a. Complete the table, include the labels.
b. Sketch a graph on your paper, include the
labels.
c. What is the rate of change? How does
the rate of change relate to the problem?
0
1
2
3
d. What is the fixed number? How does the
fixed number relate to the graph?
e. What is the function rule?
f. Graph this function on the calculator and
identify a good viewing window for this
problem. Sketch on your paper:
X min=
X max=
Y min=
Y max=
© Copyright 2006 – LeAnn Barr
g. What is the independent variable?
h. What is the domain?
i. What is the dependent variable?
j. What is the range?
k. How much would it cost if the trip was 30 miles?
l. How much would it cost if the trip was 12 miles?
m. How much would it cost if the trip was 7 miles?
n. How many miles was the trip if the cost was $28?
o. How many miles was the trip if the cost was $32.50?
Algebra I - TEXAS Style
16
Functions Activity “Taxi”
Lesson 2-3
Multiple choice:
2. Which of the following is not a functional relationship?
A. {(-3,2), (-4,2), (-5,3)}
B.
C.
D.
3. Lupe’s pattern has a function rule of L(x)=3x-1. Find the range if the domain is {2,4,6}.
A.
B.
C.
D.
{6,12,18}
{5,11,17}
{-6,-12,-18}
{4,6,8}
© Copyright 2006 – LeAnn Barr
4. If the following figure is reflected across the x-axis, what
would be the coordinates of ABC?
A.
B.
C.
D.
(-4,-4), (-1,-2), (-3,-1)
(4,4), (1,2), (3,1)
(-4,-4), (-2,-1), (-1,-3)
(4,4), (2,1),(1,3)
5. Patsy is publishing a book. Print Shop Plus charges a setup fee of $50 and $0.08 per page.
Her book contains 96 pages. Which of the following is the independent variable?
A.
B.
C.
D.
Cost per book
Setup fee
Total cost
Number of pages
Algebra I - TEXAS Style
17
Functions – More Domain and Range
Lesson 2-4
Warmup:
1. Use the table to answer each question.
a. rate?
b. Fixed number?
c. F(n)=
d. f(5)=
e. F(x)=59, find x
Term#
1
2
3
Value
3
7
11
2. Bob is going to paint his house. He rented a paint sprayer for $39 a day and
he spent $400 on paint.
a. Complete a table using the domain of {0,1,2,3} and total amount spent on
paint.
b. Rate?
Fixed number?
F(n)=
c. Domain and range
d. Write a depend statement.
3. Draw an example of a graph that is a function.
4. Draw an example of a graph that is not a function.
Domain represents the x variable.
© Copyright 2006 – LeAnn Barr
Range represents the y variable.
Find the range of the function with the given domain:
Example 1: a(x) = 2x – 1, D = {-1,0,1}
Use a table to find
the missing values.
Try these:
1. f(x) = 3x + 1, D = {-2,0,2}
a(-1)
a(0)
a(1)
x
-1
0
1
2x - 1
2(-1) – 1
2(0) - 1
2(1) - 1
y
-3
-1
1
2. h(x) = -2x – 2, D = {-2,-1,0}
Algebra I - TEXAS Style
18
Functions – More Domain and Range
Lesson 2-4
Find the domain of the function with the given range:
Example 2: g(x) =
1
x + 1 = −1
2
−1 −1
1
x = −2
2
1
2 • x = −2 • 2
2
x = −4
Try these:
1
x + 1 , R = {-1,3,5}
2
1
x +1 = 3
2
−1 −1
1
x=2
2
1
2• x = 2•2
2
x=4
1
x +1 = 5
2
−1 −1
1
x=4
2
1
2• x = 4•2
2
x =8
3. h(x) = 3x + 2, R = {-4,2,11}
This time we solve an
equation by setting
the function equal to
the numbers in the
range.
4. c(x) = 2.50x – 1, R = {1,4,9}
Verify your answers using your calculator. TI-83 or TI 83 plus.
1. Enter the function in y=.
2. Compare your answers with those in the table (2nd, Graph).
TAKS Practice Questions:
© Copyright 2006 – LeAnn Barr
5. Find the range of the function g(x) = 2x – 3, if the domain is {-2,0,4}.
A.
B.
C.
D.
{1,-3,11}
{1,-3,5}
{-7,-3,5}
{-7,-3,11}
6. Erica was given the following formula for finding the cost of an ad in the
newspaper. C(x) = 0.5x+3, where x represents the number of characters
and C(x) is the total cost. If the total cost of several ads were represented
by the range {15,17,20}, what was the domain and what did it represent?
A. {24,28,34} characters
B. {$24,$28,$34}
C. {$10.50,$11.50,$13.00}
D. {10.5,11.5,13} characters
Algebra I - TEXAS Style
19
Functions
Airplane Problems
Lesson 2-5
Warmup:
1. Solve each:
a.
1
x+4=9
3
b. − 3 x + 11 = −22
2. f(x)=x+9, find the range if the domain is {-1,0,2}.
3. I-Scream Ice Cream Shoppe charges $0.75 for the first scoop of ice cream and
$0.50 for each additional scoop.
a. Complete the table:
b. What is the rate of change?
c. What is the fixed number?
d. Write the function rule.
e. C(4) = ? How does this relate to
the problem?
Example 1: Samantha is approaching the airport in her airplane at 3500 ft. The air
traffic controller tells her to descend to the runway at 500 ft per minute.
a. Complete the table.
b. What is the rate of
change and what
does it have to do with this problem? -500,
© Copyright 2006 – LeAnn Barr
because the plane is descending at 500 ft/min.
c. What is the fixed number and what does it
have to do with this problem? 3500,
because this is the starting altitude and you
subtract from this number each time.
d. What is the function rule? F(x) = 3500 – 500x
e. F(5) = 3500 – 500(5) = 1000
f. F(10) = 3500 – 500(10) = -1500. This value does not make sense for the problem so
it is not part of the domain.
g. What is the Domain and what does it represent? D = {0 < x < 7} minutes,
because at 7 minutes she has landed the airplane.
h. What is the Range and what does it represent? R = {0 < y < 3500} feet.
Algebra I - TEXAS Style
20
Functions
Airplane Problems
Lesson 2-5
Example 2: Flyboy Fred is flying his plane at an altitude of 1200 ft. and is told by the
control tower to be at an altitude of 2000 ft in 4 min.
a. Complete the table
b. What is Fred’s average rate
of ascent? Fred must ascend
800 feet in four minutes.
2000 − 1200
= 200 ft / min
4
c. What is the y-intercept and what does it have to do with this problem? This
is Fred’s starting altitude.
d. What is the function rule? F(x) = 200x + 1200
e. Write two algebraic representations for the function rule.
The sum of 200 times a number and 1200.
1200 more than 200 times a number.
f. What is the domain and range for this problem? D = 0 to 4 min, R = 1200 to
© Copyright 2006 – LeAnn Barr
2000 ft
Example 3, TAKS Practice:
A rectangular room has an area of 200 sq ft. A model is created using
both dimensions. What is the area of the model?
A.
B.
C.
D.
800 sq ft
400 sq ft
50 sq ft
12.5 sq ft
Since this is area you have two dimensions,
length and width to consider. As a result, you will
multiply ¼ two times, once for the length and once
for the width.
1 1
⋅ ⋅ 200 = 12.5
4 4
The correct answer is D. 125 sq ft.
Algebra I - TEXAS Style
21
1
th of
4
Functions and Equations With Variables on Both Sides
Lesson 2-6
Example 1: Ron is planning to publish a book of his music.
He has contacted two publishers, and wants the best deal.
Big Band Publishers charges a set up fee of $250 plus $3 per copy.
Music and More charges a set up fee of $150 plus $5 per copy.
© Copyright 2006 – LeAnn Barr
a. Complete a table:
b. Big Band Publishers
What is the rate of change and how does it relate to this problem? $3, this
is the change per book.
What is the y-intercept and how does it relate to this problem? $250, this is
the setup fee.
What is the function rule for Big Band Publishers? B(x) = 3x+250
c. Music and More
What is the rate of change and how does it relate to this problem? $5, this is
the change per book.
What is the y-intercept and how does it relate to this problem? $150, this is
the setup fee.
What is the function rule for Music and More? M(x) = 5x+150
Algebra I - TEXAS Style
22
Functions and Equations With Variables on Both Sides
Lesson 2-6
d. Find the cost of publishing 40 books from both companies.
Big Band B(40) = 3(40) + 250 = 370, Music and More: M(40) = 5(40) + 150 = 350
e. How many books can be published for $580 from both companies?
Big Band Publishing
Music and More
e. Who has the best deal? You can’t tell the exact value yet, but for low numbers use Music and
More and for high numbers use Big Band.
f. Write an equation to represent the point where both
companies charge the same amount for the same number of
books, and solve it. First get all the x’s on one side, by subtracting 3x
from both sides. Then move the numbers with no variables to the opposite
side, by subtracting 150 from each side. Lastly, divide to get your answer.
At 50 books both companies charge the same amount. Verify with your
calculator by looking in the table.
© Copyright 2006 – LeAnn Barr
g. Graph the two lines on your calculator and find an appropriate viewing window
to show the intersection of these lines.
h. When is each company a better deal?
Music and More for up to 50 copies, D = {0 < x < 50},
Big Band for more than 50 copies, D = {x > 50}
Try these equations, follow the steps from the example above.
1. 4x + 7 = 2x – 19
2. 7x – 6 = 3x - 30
Algebra I - TEXAS Style
23
Functions – Variables on Both Sides
Lesson 2-7
Warmup
1) This chart represents the number of students who could attend a play and the
total cost for the trip, including the fee to use the bus.
X
Y
a.
b.
c.
d.
e.
15
306.25
20
325
38
392.50
Label the table. (Turn it the other way if you need to)
Sketch a graph.
What is the rate of change and how does it relate to the problem?
What is the y-intercept and how does it relate to the problem?
F(x)= Explain the equation in a complete sentence.
2) The Golf Club has two options for playing golf.
Option 1: $50 membership fee plus $5 per game.
Warmup
Option 2: No membership fee and $15 per game.
a. Write the function rule for option 1. M(n)=
b. Write the function rule for option 2. N(n)=
c. Write the equation for when the two options are the same and solve it.
d. Which is a better deal and when?
© Copyright 2006 – LeAnn Barr
3) Solve each:
a. 5 x + 2 = 3x + 12
b. − 5 x = 2 x + 63
c. 4 x − ( 2 x − 3) = 4 x − 11
TAKS Practice:
1) A scuba diver has been exploring a sunken ship 100 ft below the surface of
the ocean. He will begin swimming to the surface at a rate of 5 ft/sec. Which
equation shows when he will reach the surface?
A.
B.
C.
D.
5x – 100 = 0
5x + 100 = 0
100 – 5x = 0
-5x – 100 = 0
Algebra I - TEXAS Style
24
Functions – Variables on Both Sides
Lesson 2-7
2) Which of the following situations could represent this equation:
F(x) = 5x + 10
A. A phone on an airplane charges $5 for the first minute and $10 for
each additional minute.
B. Danny has $10 and plans to save an additional $5 per week.
C. Sue bought a $10 pair of shoes and paid 5% in sales tax.
D. Bob paid a membership fee of $5 so he can play golf at $10 per game.
A.
B.
C.
D.
Videos N More
50
45
40
35
30
Total Cost
3) Videos N More rents DVDs for $4.00 each for
non-members. Members can rent the DVDs for
$2.50 each with a $15.00 membership fee. How
many videos do you have to rent to begin saving
money?
25
20
5
10
40
100
15
10
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Number of DVDs
4) Which of the following relations does not represent a function?
© Copyright 2006 – LeAnn Barr
A.
B.
C.
5) Which of the following represents the solution to this equation?
2x – 6 = 10x - 36
A.
B.
C.
D.
8.5
-3.5
3.75
-8.5
Algebra I - TEXAS Style
25
D.
Functions – Using Coordinates
Lesson 2-8
Warmup:
The total cost of a field trip includes a ticket for each student and the
cost of the bus for all students.
a. Label the independent and dependent variables.
b. Write a depend statement.
c. What is the domain?
d. What is the rate that is charged for 1 student? (The
cost of a ticket)
e. What is the bus fee?
f. Write the function rule.
# of
students
Total
cost
5
6
7
8
9
10
15
20
n
55
70
85
100
Example:
The Theater class is planning to attend a play. Mrs. Windham was
given the number of students attending from other classes and the
total cost, including the fee to use the bus. 15 students attended and
the total cost was $306.25, written as (15,306.25) where the x
represents the number of students and the y represents the total
cost. (20,325) and (38,392.50) were some other numbers of students
and total cost.
Number of
Students
Process
Total
cost
15
3.75(15)+250
306.25
5
20
3.75(20)+250
325.00
18
38
3.75(38)+250
392.50
© Copyright 2006 – LeAnn Barr
a. Put these values in a table:
b. Sketch a graph:
c. What is the rate of change and how does it relate to the problem? The change in
price for 5 students is $18.75. So $3.75 is the cost of 1 student ticket. (Check this with the
next set of values:
is also 3.75)
Algebra I - TEXAS Style
26
18.75
67.50
Functions – Using Coordinates
Lesson 2-8
d. What is the y-intercept and how does it relate to the problem?
Solve an equation using the process column
The y-intercept = 250 and it is the cost of the bus.
e. Write the function rule and explain the equation in a complete sentence. f(x)=3.75x+250,
the tickets cost $3.75 each and the bus fee is $250 for entire trip.
f. What does the point (38,392.50) mean for this problem? 38 students attending the play
cost a total of $392.50
g. Find the value of (25,y), and explain: 3.75(25)+250=343.75 25 students attending the play
cost a total of $343.75.
Try these:
h. Find the value of (30,y) and explain:
i. Find the value of (40,y) and explain:
j. Find the value of (x,347.50) and explain: (Be careful, you will have to solve an
equation)
© Copyright 2006 – LeAnn Barr
k. Find the value of (x,407.50) and explain:
Try this one: Joe’s friends have bought daisies and a vase for their
girlfriends. Bill bought 2 daisies with the vase and spent $5.50
represented by (2, 5.50). George’s could be represented by (5, 7.75)
and Robert’s could be represented by (12, 13.00).
a.
b.
c.
d.
e.
Put the coordinates in a table. Label the columns.
What is the rate of change and what does it represent?
What is the y-intercept and what does it represent?
What is the equation that represents this problem?
If Joe’s coordinate was (x, 15.25), solve this problem and explain what it
means.
f. Eric bought twice as many daisies as Robert. How much did he spend?
Algebra I - TEXAS Style
27
Graphing Linear Equations
3-1 Scatter Plots and Correlation Practice …………………………page 30
a.
b.
c.
d.
e.
Plotting scatter plots on the calculator
Finding the line of best fit.
Identifying correlation from graphs and words.
Workbook Set 3-1
TEKS a3, b1D, b2C, b2D
New Numbered TEKS a3, A.1D, A.2C, A.2D
3-2 Direct Variation ……………………………………………………..page 32
a.
b.
c.
d.
Vocabulary for Direct Variation
Constant of Variation and Function Rules
Workbook Set 3-2 A, 3-2 B, 3-2 C
TEKS same as above plus b1A, b1B, b1C, b1E, b2B, b2D, b3A, b3B, b4A, b4B,
c1B, c1C, c2A, c2G, c3A, c3B, c3C
New Numbered TEKS A.1A, A.1B, A.1C, A.1E, A.2B, A.3A, A.3B, A.4A, A.4B, A.5B,
A.5C, A.6A, A.6G, A.7A, A.7B, A.7C
3-3 Graphing Linear Equations – Slope …………………………….page 34
a.
b.
c.
d.
e.
Finding Slope and Graphing using slope.
Writing equations of lines.
Graphing and Comparing to the parent function y=x.
Workbook Set 3-3 Activity, 3-3 B
TEKS same as above plus c2B, c2D, c2E
© Copyright 2006 – LeAnn Barr
New Numbered TEKS A.6B, A.6D, A.6E
3-4 Graphing Linear Equations – More Slope ……………………..page 36
a. Graphing a line using slope and a point
b. Workbook Set 3-4
c. TEKS same as above
3-5 Graphing Linear Equations – y=mx+b …………………………page 38
a.
b.
c.
d.
Graphing equations of lines
Identifying slope and y-intercept.
Workbook Set 3-5 A and Set 3-5 B Distance and Time
TEKS same as above plus c2F
New Numbered TEKS A.6F
Algebra I - TEXAS Style
28
3-6 Graphing Linear Equations – Domain and Range …………page 40
a.
b.
c.
d.
Using domain and range.
Vertical shifting of Lines.
Workbook Set 3-6 A, and 3-6 B
TEKS same as above plus c2C
New Numbered TEKS A.6C
3-7 Parallel and Perpendicular Lines ……………………………. page 42
a. Parallel and Perpendicular lines
b. Workbook Set 3-7 Activity and 3-7 B
c. TEKS same as above
3-8 Distance Between Two Points and Midpoint ……………….page 44
TEKS 8.7, 8.9, 8.14, and 8.15
3-9 Standard Form ……………………………………………………page 46
a. Workbook Sets 3-9A, 3-9B, 3-9C, Chapter 3 Review
b. TEKS same as above
© Copyright 2006 – LeAnn Barr
TAKS Practice …………………………………………………………page 48
Algebra I - TEXAS Style
29
Graphing – Correlations and Scatter Plots
Lesson 3-1
Types of Correlation:
Identify the Independent and Dependent variables. Write a relationship statement. Sketch
the graph and identify the correlation.
Example 1: The size of a window and the amount of fabric needed to make curtains.
Independent Variable: Size of the window
Dependent Variable: Square yards of fabric.
Relationship statement: The square yards of fabric need for
curtains depends on the size of the window.
Correlation: Positive.
Example 2: The amount spent on a boat and the number of fish caught.
Independent Variable: None.
Dependent Variable: None.
© Copyright 2006 – LeAnn Barr
Relationship Statement: None.
Correlation: None
This problem has two possible graphs, both have no correlaton.
Example 3: Number of gumballs in the gumball machine and the number of days since it
has been filled.
Independent Variable: Number of days.
Dependent Variable: Number of gumballs.
Relationship Statement: The number of gumballs in the gumball
machine depends on the number of days since it has been filled.
Correlation: Negative.
Algebra I - TEXAS Style
30
Graphing – Correlations and Scatter Plots
Lesson 3-1
To draw a scatter plot:
1. STAT, EDIT, (enter) Put your values in L1 and L2.
2. 2nd, Y=, (STATPLOT) to turn on Plot 1 and check values.
3. WINDOW
a. Xmin (smaller than the smallest L1 value)
b. Xmax (larger than the largest L1 value)
c. Ymin (smaller than the smallest L2 value)
d. Ymax (larger than the largest L2 value)
4. GRAPH
To find the line of best fit:
5.
6.
7.
8.
9.
STAT, right arrow to CALC
Choose ax+b, enter.
Write your equation using the numbers it gives you.
Y=, enter your equation
GRAPH to see it
Example 4: Put the data in the calculator using the steps from above. Then decide if it shows
positive, negative or no correlation. Find the line of best fit. (Make sure your calculator screen
matches the screen shots for each step below.)
X
Y
1
3
2
5
3
6
4
9
5
11
6
15
7
16
X – Number of hours at the booth selling crafts, Y – Total Number of Sales
© Copyright 2006 – LeAnn Barr
Step 1:
Step 3:
Step 6:
Step 2:
Step 4: positive correlation
Step 7: y = 2.17x + 0.5
Step 8:
Algebra I - TEXAS Style
31
Step 5:
Step 9:
8
17
9
20
Graphing – Direct Variation
Lesson 3-2
The value of dimes varies directly with the number of dimes counted.
1. Complete the table:
Number
of Dimes
Direct Variation
always goes
through the origin.
Value
0
1
2
.10(0)
.10(1)
.10(2)
n
.10n
0
.10
.20
Direct Variation
is always linear.
If a function has
direct variation then
the rate of change
is sometimes called
the constant of
variation.
2. Draw a graph:
3. a. What is the rate of change? 0.10
b. What is the y-intercept? 0
© Copyright 2006 – LeAnn Barr
c. What is the function rule? y = 0.10x + 0 or y = 0.10x
4. If I had 35 dimes, how much money would I have? y = 0.10(35) = $3.50
5. If I had $7.90 in dimes, how many dimes did I have? 7.9 = 0.10 x
0.10 0.10
79 = x
79 dimes
6. What does the ordered pair (5, 0.50) mean for this problem? 5 dimes are worth $.50
7. For the equation y=3x, answer each question:
a. rate ? 3
b. y-intercept ? 0
c. Is this direct variation? yes
8. For the equation y=2x-4, answer each question:
a. rate ? 2
b. y-intercept ? -4
c. Is this direct variation? no
Algebra I - TEXAS Style
32
Graphing – Direct Variation
Lesson 3-2
Try this one:
1. George is making picture frames. The total amount of wood used varies directly with the
number of frames built. Each frame uses 3 ft of wood.
a. Create a table; include the labels.
b. Sketch a graph.
c. If George builds 4 frames, how much wood will he need?
d. If George used 21 ft of wood, how many frames did he build?
e. What does the ordered pair (x,33) mean for this situation? Find the value of x.
f.
What is the constant of variation?
g. What is the y-intercept?
h. What is the function rule?
2. Direct variation always goes through the _________, which has coordinates of ( _ , _ )?
3. If something is linear, it has the same ______ ___ _____ for every set of coordinates.
4. If y varies directly with x, with is the other name for rate of change?
© Copyright 2006 – LeAnn Barr
5. For the equation y = 4x – 2, answer each question:
a. rate of change?
b. y-intercept?
c. Is this direct variation?
d. (x,18), x = ?
e. (10,y), y = ?
f. (x,42), x = ?
6. For the equation y = 7.5x, answer each question:
a. rate of change?
b. y-intercept?
c. Is this direct variation?
d. (5,y), y = ?
e. (x,60), x = ?
f. (10,y), y = ?
7. Which of the following graphs represent direct
variation?
Algebra I - TEXAS Style
33
Graphing Linear Equations – Slope
Example 1:
Lesson 3-3
Slope is the
same as rate
of change.
Put the
coordinates in a
table, to find the
slope.
What if you count
from dot to dot and
record as
rise ⇑⇓
or
.
run
⇔
A. From your table what is the slope?
1
1
© Copyright 2006 – LeAnn Barr
y
-1
2
5
3
1
B. When you count from dot to dot and record as
the graph.
x
0
1
2
3
1
rise
, what is the slope? Look at the red lines on
run
Notice: It was the same for either method.
C. What is the y-intercept for the graph above? It crosses the y-axis at -1, so the y-intercept is -1.
D. What is the function rule for the graph above? Put the slope before the x and the y-intercept at
the end, so your equation is Y = 3x – 1, compare this to the function rule that you get from using a
table.
x
y
3(0)-1
0
-1
1
3
3(1)-1
1
2
1
3(2)-1
2
5
3
Algebra I - TEXAS Style
34
3
3
Graphing Linear Equations – Slope
Example 2:
A. Create a table to find the slope. Use points from
the line.
2
2
x
0
2
4
y
1
0
-1
-1
-1
B. Count to find slope. Look at the red drawn on the
graph. It has a rise of -1 (falls 1) and runs 2. So the
−1
. (Notice it is going down from left to
slope is
2
right so it is negative.)
C. What is the y-intercept? It crosses the y axis at 1.
D. What is the function rule? y = - ½ x + 1
Try these:
1. Answer each question about the graph:
A. Create a table to find the slope.
© Copyright 2006 – LeAnn Barr
B. Count to find the slope.
C. What is the y-intercept?
D. What is the function rule?
2. Answer each question about the graph:
A. Write some specific points to use for this line.
B. Create a table to find the slope.
C. Count to find the slope.
D. What is the y-intercept?
E. What is the function rule?
Algebra I - TEXAS Style
35
Lesson 3-3
Graphing Linear Equations – Slope
Lesson 3-4
Example 1: Sketch the graph of a line with the given slope through the
given point. Then write the equation for the line.
Step 3
(2,3)
Step 1
slope (m) =
Run
+2
1
2
Put your point
in the table.
Remember:
Rise
Run
x
2
4
6
Put the
points on
the graph.
Step 4
y
3
4
5
Rise
+1
Draw the
line.
Step 2
Find another
point using the
slope.
© Copyright 2006 – LeAnn Barr
Here is a way to check
yourself. Count the
slope from one point
to another, graphing
each point as you go.
Then draw your line.
Notice the red lines.
What is the slope (m)? ½ ,
y-intercept (b)?
2,
equation?
y= ½x+2
Try this one:
1. Through (1, -1) with m =___
2. Make a fraction by putting a 1 in the denominator (bottom).
1
A.
B.
C.
D.
Draw a table and find one more point.
Plot the points and draw the line on a graph the size of the example.
Show the slope along the line and extend to the edges of the graph.
Write the slope, y-intercept , and equation of the line.
Algebra I - TEXAS Style
36
Graphing Linear Equations – Slope
Lesson 3-4
Example 2: Find the slope between each set of coordinates.
a. (3,2) and (1,0) Run
-2
x
3
1
y
2
5
x
-3
-2
y
7
4
x
1
4
y
3
3
x
2
2
y
-4
5
Slope = 3
Rise
-2
+3
Leave it improper.
Use a Table.
Use a Table.
b. (-3,7) and (-2,4)
1
c. (1,3) and (4,3)
3
d. (2,-4) and (2,5)
0
-3
m = -3 = -3
1
0
m = 0 = no rise
3
run
This is a
horizontal line.
9
m = 9 = rise
0
no run
This is a vertical
line.
Try these:
© Copyright 2006 – LeAnn Barr
Find the slope between each set of coordinates.
2. (4,5) and (2,6)
3. (-3,5) and (1,-1)
4. (1,2) and (1,-3)
5. y = 3x – 2
6. y = - ½ x + 7
7. y = ⅜x - 4
8.
9.
10.
What is the slope of each line?
Algebra I - TEXAS Style
37
Graphing – Slope Intercept Form
Lesson 3-5
Example 1: Use the equation y =
Remember the
slope is the rate
of change.
answer each question.
The slope is the
coefficient of x.
(The number in
front of x).
a.
What is the slope? ⅓
b.
What is the y-intercept? 2
c.
Complete the table to find two more
points. (Use your calculator: y= and
table) Only record integers, skip the
decimal answers.
x
-3
0
3
© Copyright 2006 – LeAnn Barr
The y-intercept is
where the graph will
cross the y-axis.
1
x + 2 to
3
d.
y
1
2
3
Plot your points on the graph and draw
your line.
Try this and
compare. Slope is
RISE
. Plot the y
RUN
intercept on the y
axix, then count up
and over, plotting
other points as you
go.
Try These:
Make a table that contains at least 3 points. Identify the slope (m) and y-intercept (b) and
graph.
1
1. y = -3x – 2
2. y = x − 4
3. y = 4x
2
2
4. y = -4
5. y = x
6. x = 2
5
7. Do any of the above questions represent direct variation?
Algebra I - TEXAS Style
38
Graphing – Slope Intercept Form
Lesson 3-5
Write the equations using the given information.
Example 2: From the graph at the right:
Step 1: Identify the y – intercept. b = -1
Step 2: Find the slope: Locate points on exact corners.
(0,-1) and (3,-5) are two from this graph. Count from one
point to the next, down 4 and right 3, then record.
Rise − 4
=
3
Run
Or you can use a table, just put the two points in the table and find the
slope.
x
y
0
-1
3
-4
3
-5
−4
y
=
x −1
Step 3: Write the equation:
3
Example 3: Write an equation of the line through points (4,1) and (2,4).
Use a table, just like you did for a pattern or function.
x
4
© Copyright 2006 – LeAnn Barr
-2
2
y
1
−3
(4) + b = 1
2
− 6+ b =1
b=7
−3
(2) + 7
2
3
4
−3
So the equation is y = 2 x + 7
Example 4: Write an equation of the line through (2,-5) with a slope of −
Use a table like the one above. You already know the slope.
x
2
1
− (2) + b = −5
2
− 1 + b = −5
b = −4
y
-5
1
So the equation is y = − 2 x − 4
Example 5: Using a slope of 2 and y-intercept of 6.
This equation is just y = 2x + 6.
Algebra I - TEXAS Style
39
1
.
2
Graphing Linear Equations using Domain
Lesson 3-6
Warm-up:
1. Find the range of y = 3x-2, if the domain is {-2,-1,0}.
2. State both endpoints of the graph below.
3. Fill in the blanks for the domain: { ____ < x < ____}.
4. Fill in the blanks for the range: { ____ < y < ____}.
Example 1:
Let’s draw the graph of
a geometric shape.
x
3
x+2,
2
but we don’t want the entire line,
just the domain of {-2<x<2}
Our first equation is y =
© Copyright 2006 – LeAnn Barr
Now plot
the points
on the
graph and
connect
the dots.
Find the
ends
using the
table.
-2
2
3
x+2
2
3
(−2) + 2
2
3
(2) + 2
2
Check your
answers using
your calculator.
Put the equation
in y= , then go to
the table: (2nd
GRAPH)
Do the same for the equations
−1
y=
x − 2 D={-2<x<2}
2
and
x = 2 D={-3<y<5}.
x
-2
2
−1
x−2
2
−1
(−2) − 2
2
y
−1
(2) − 2
2
-3
-1
x
2
2
X=2
y
-3
5
X=2 for all the values of y. so
put -3 and 5 in the y column and
2 in the x column.
Algebra I - TEXAS Style
40
y
-1
5
Graphing Linear Equations using Domain
Lesson 3-6
Now let’s write equations
for the segments in a
triangle.
First find the endpoints
of each line, and put
them in a table.
Use the table and graph to find
the equation. Sometimes all of
your information can come from
the graph, otherwise use a table.
x
-4
3
x
2
© Copyright 2006 – LeAnn Barr
3
x
-4
2
red line
(-4,-3) & (3,4)
1(-4) + 1
1(3) + 1
y
green line
(2,-3) & (3,4)
7(2) − 17
7(3) − 17
y
-3
4
-3
4
y
Blue line
(-4,-3) & (2,-3)
0(-4) - 3
0(2) - 3
-3
-3
The red line has a slope of 1 and y-intercept of 1,
so the equation is y = 1x + 1 or y = x + 1. The
domain is {-4 < x < 3} (left sides of the table) and
the range is {-3 < y < 4} (right side of the table.
The green line has a slope of 7 and a y-intercept
of -17, so the equation is y = 7x -17.
D = {2 < x < 3} and R = {-3 < y < 4}.
The blue line has a slope of 0 (horizontal line) and
a y-intercept of -3, so the equation is y = -3.
D = {-4 < x < 2} and R = {-3}
Try these:
1. Find the endpoints of the line y = 3x – 1 for the domain {-1 < x < 2}.
2. Write the equation of the segment at the right, include the domain.
Algebra I - TEXAS Style
41
Graphing – Parallel and Perpendicular
Lesson 3-7
Example 1:
Line:
Look at the parallel lines
below.
m=
y=
2
5
2
x , blue line.
5
b= 0
x
-5
0
5
What is the same in both
equations?
Line:
m=
2
5
y=
2
x + 2 , green line.
5
b= 0
x
-5
0
5
Both lines have
2
a slope of .
5
© Copyright 2006 – LeAnn Barr
y
0
2
4
Try these:
Match equations that are parallel to each other.
1. y = 4 x − 1
Two lines are
parallel if their
slopes are the
same.
y
-2
0
2
2. y =
2
x +1
3
3. y = x − 5
−1
x −1
3
5. x = 2
4. y =
6. y = 3
A. y =
2
x
3
B. y = 2
C. y =
−1
x+3
3
D. x = 5
E. y = 4 x + 3
F. y = x + 2
7. Write the equation of the line that is parallel to
3
y = x − 1 and passes through (-2,0). (hint: use
2
the slope of the line)
Algebra I - TEXAS Style
42
Graphing – Parallel and Perpendicular
Look at the graphs of these
two equations.
Example 2:
Line: y = 3x + 5 , blue line.
3
m=
b= 5
1
x
y
-3
-4
-2
-1
0
5
Line:
m=
y=
−1
3
Lesson 3-7
How do the slopes
compare?
−1
x − 5 , green line.
3
b= -5
x
-3
0
3
y
-4
-5
-6
The slopes of
perpendicular lines
are opposites and
flipped over. This
is called a negative
reciprocal.
Try these:
Matching the equations that are perpendicular.
© Copyright 2006 – LeAnn Barr
8. y =
−3
x−2
2
A. y = −3x − 2
1
9. y = x + 2
3
B. y = −
10. y = −2 x + 3
C. y =
11. y = 4 x − 1
D. y = 3
12. x = 3
E. y =
13. y = −1
F. x = 1
1
x+2
4
1
x+2
2
2
x +1
3
14. Write the equation of a line that is perpendicular to y =
−1
x + 1 and has the same y-intercept.
2
Algebra I - TEXAS Style
43
Graphing – Distance Between Points
Example 1:
Lesson 3-8
Sometimes we need to find the distance
between two points.
To find the distance between (-5,1) and (7,6),
first plot the points. Then draw a right triangle,
see the red and green line.
Now find the length of
each leg of the right
triangle.
Green = 7-(-5)=12
Red = 6-1=5
Now use the Pythagorean
Theorem. a 2 + b 2 = c 2
5 2 + 12 2 = c 2
25 + 144 = c 2
169 = c 2
© Copyright 2006 – LeAnn Barr
169 = c
13 = c
Now lets work backwards to develop a formula.
c2 = a2 + b2
Start with the Pythagorean Theorem.
Substitute in the values for our problem.
c 2 = 12 2 + 5 2
Now substitute these values:
c 2 = (7 − ( −5)) 2 + (6 − 1) 2
Green = 7-(-5)
Red = 6 – 1
c = (7 − ( −5)) 2 + (6 − 1) 2
Square root both sides.
c = ( x2 − x1 ) 2 + ( y 2 − y1 ) 2
Since the first x ( x1 )= -5 and second x ( x2 )=7
And the first y ( y1 )= 1 and second y( y 2 )=6, substitute these into the equation.
Now look at your formula chart to compare the two formulas.
Example 2: Now use the formula to find the distance between (4,1) and (7,5).
d = (7 − 4) 2 + (5 − 1) 2
= (3) 2 + ( 4) 2
= 9 + 16
= 25
=5
Algebra I - TEXAS Style
44
Graphing – Distance Between Points
Lesson 3-8
Try these:
1. (4,2) and (1,-2)
Midpoint formula: (
2. (3,-1) and (-2,6) (round to the nearest tenth)
( x1 + x2 ) ( y1 + y 2 )
,
)
2
2
I just average the x’s and y’s.
Example 3: Find the midpoint of (-2,4) and (-8,12)
(−2 + −8) − 10
=
= −5
2
2
(4 + 12) 16
y=
=
=8
2
2
x=
so the midpoint is (-5,8).
© Copyright 2006 – LeAnn Barr
Now find the midpoint of these points:
3. (6,1) and (10,5)
4. (-1,4) and (7,-8)
5. Write the equation of the line that passes through
the midpoint of the segment in the graph and has a
slope of 2.
6. Are the two lines perpendicular? Why or why not?
Algebra I - TEXAS Style
45
Graphing – Standard Form
Lesson 3-9
Standard form has all the
variables on one side of the
equation like 2 x + 4 y = 8
Sometimes the equation
is in standard form.
Example:
Steps to get y by itself.
2x + 4 y = 8
• Subtract the x’s
from both sides.
• Divide everything
in the equation by
the coefficient of
y.
• Rewrite.
• Graph like always.
− 2x
− 2x
4 y = −2 x + 8
4 y − 2x 8
=
+
4
4
4
y=
−1
x+2
2
A. What is the slope for the example? -½
B. What is the y-intercept? 2
© Copyright 2006 – LeAnn Barr
C. Complete a table and graph.
x
-2
0
2
Try this one:
‰ Subtract 3x from
both sides.
‰ Divide everything
by –2.
‰ Rewrite.
‰ Simplify and graph.
y
3
2
1
1.
3x − 2 y = 8
Algebra I - TEXAS Style
46
Identify m and b, create a table,
then graph.
Graphing – Standard Form
Lesson 3-9
Try these: Solve each for y, identify m and b, create a table, then graph.
2. 3x - 2y = -6
3. 2x + 3y = 12
Multiple choice:
4. Which equation represents the graph at the right?
First, write the
equation for
the graph in y=
form.
Then
solve each
choice
for y, to
see which
one
matches.
A.
x – 3y = -3
B.
x + 3y = 3
C.
x – 3y = 3
D.
x + 3y = -3
© Copyright 2006 – LeAnn Barr
5. Which equation represents the graph at the right?
A.
B.
C.
D.
3x + 2y = -2
3x + 2y = 2
3x – 2y = -2
3x – 2y = 2
6. Which of the following equations represents the line that passes through (-4,-2) and (0,1)?
A.
B.
C.
D.
3x – 4y = 4
3x – 4y = -4
3x + 4y = 4
3x + 4y = -4
Algebra I - TEXAS Style
47
TAKS Practice
1. A triangle has an area of 45 cm2. If
the base and height are both tripled,
what is the new area?
A.
B.
C.
D.
9 cm2
48 cm2
135 cm2
405 cm2
x 1
y 2
2. A package of colored paper contains
four colors of paper, 20% pink, 30%
light blue, 10% light green, and 20
sheets of canary yellow. How many
pink sheets are in the package?
A.
B.
C.
D.
20
10
50
5
3. Which of the following does not
represent a function?
© Copyright 2006 – LeAnn Barr
A. {(0, 1), (1, 2), (2, 3), (3, 4)}
B. {(-5, 2), (-4, 1), (-3, 0), (-2, -1)}
C.
D.
4. Which of the following function
rules best represent the values in this
table?
A.
B.
C.
D.
2
5
3 4 5
8 11 14
f(x)=3x-1
f(x)=3x+1
f(x)=2x+2
f(x)=2x+1
5. Lucy’s pattern has a function rule of
L(x)=2x+3. Find the range of
Lucy’s pattern if the domain is
{-1, 0, 1}.
A.
B.
C.
D.
{-2, -1.5, -1}
{5, 7, 9}
{1, 3, 5}
{2, 3, 4}
6. A taxi charges a flat fee of $3 plus
$1.25 per mile. What is the
independent variable?
A.
B.
C.
D.
taxi
$1.25
cost
miles
7. Grant has grades of 85, 92, 90, 81,
92 on tests in Algebra. Which
measure of central tendency is most
affected by changing the 85 to 91?
A.
B.
C.
D.
Algebra I - TEXAS Style
48
mean
median
mode
range
Inequalities
4-1
Inequalities with One – Variable ………………………………..page 50
a. Solving and Graphing One-Variable Inequalities
b. Workbook Set 4-1
c. TEKS - b1C, b1D, c3A, c3B, c3C
New Numbered TEKS – A.1C, A.1D, A.7A, A.7B, A.7C
4-2 Inequalities with Two – Variables ………………………………page 52
a. Solving and Graphing Two-Variable Inequalities
b. Workbook Set 4-2
c. TEKS – b1E, b2A, b2D
New Numbered TEKS – A.1E, A.2A, A.2D
4-3
Inequalities Application Problems ……………………………..page 54
a. Application problems using Inequalities.
b. Workbook Set 4-3A,4-3B,4-3C
c. TEKS same as above
© Copyright 2006 – LeAnn Barr
TAKS Practice …………………………………………………………….page 56
Algebra I - TEXAS Style
49
Inequalities – One Variable
Lesson 4-1
Warm-up:
1. Copy the following and write the correct comparison word in the blank: greater than or less
than.
a. 3 _________ 5
b. – 2 _________ - 6
c. 10 _________ 7
d. -4 __________ 3
2. Now copy the same problems from question 1 and use the inequality symbols > or <.
Inequality Symbols:
> means Greater Than
> means Greater Than or Equal To (notice the underline)
< means Less Than
< means Less Than or Equal To
When graphing on a number line:
> or < use an open circle and shade the appropriate side. ○
> or < use a closed circle because it includes the number (equal to).●
Example 1: x > 3
<-----------○========>
3
© Copyright 2006 – LeAnn Barr
Example 2: x < 5
<========○------------->
5
“X is greater than 3.” This one has an open circle at 3
and shaded to the right, towards greater numbers.
“X is less than 5.” This one has an
open circle at 5 and shaded to the
left, towards smaller numbers.
Example 3: x > -2
<------------●========>
-2
Example 4: x < 1
<=======●----------->
-2
“X is greater than or equal to -2.”
Draw a closed circle at -2 and shade
to the right, towards bigger numbers.
“X is less than or equal to 1.” Draw a
closed circle at 1 and shade to the left,
towards smaller numbers.
Did you notice the greater than or less than symbol
matches the end of the line you shaded towards.
Algebra I - TEXAS Style
50
Inequalities – One Variable
Lesson 4-1
Now lets see what happens when we use mathematical
operations on numbers that have an inequality.
FOLLOW ALL THE STEPS BELOW.
Start with 5 < 10.
1. Add 3 to both sides. Is it still true?
2. Subtract 9 from both sides. Is it still true?
3. Multiply by 4 on both sides. Is it still true?
4. Divide by 2 on both sides. Is it still true?
5. Add -4 to both sides. Is it still true?
6. Multiply by -2 on both sides. Is it still true?
7. What happened?
Check your answers with those in the box at the bottom of the page.
Here are some examples of solving inequalities.
Example 6:
2X-1> -15
+1 +1
2X > -14
2
2
X > -7
<----------●======>
-7
Example 8:
x
+ 7 ≤ −10
2
x
2 ⋅ ≤ −17 ⋅ 2
2
x ≤ −34
<======●--------->
-34
Example 7:
-3X-5 < 10
+5 +5
-3X < 15
-3 -3
X > -5
<---------●======>
-5
Example 9:
x
+ 2 > −3
−4
x
− 4⋅
> −3 ⋅ −4
−4
x < 12
<======○--------->
12
Try these: Solve and Graph.
1. 3x – 7 < 20
3.
2x
+ 1 ≤ −5
3
Ex 7 and 9: Solve it as
usual, except circle the
symbol when you divide
or multiply by a negative
number. Then in the
next step, reverse it.
2. -2x + 5 < 17
4.
x
−1 > 3
−5
Algebra I - TEXAS Style
51
Answer
5<10
1. 8<13, yes
2. -1<4, yes
3. -4<16, yes
4. -2<8, yes
5. -6<4, yes
6. 12<-8, NO
7. 12>-8, when
multiplying by a negative
number, the symbol must
be reversed.
© Copyright 2006 – LeAnn Barr
Example 5:
3X+2<14
-2 -2
3X<12
3 3
X<4
<======○--------->
4
Inequalities
Lesson 4-2
Now let’s graph an inequality.
Example:
2x + 3 y < 9
Step 1: Solve the inequality for y. Remember the rule:
− 2x
− 2x
change the symbol if you divide or multiply by a negative
number.
3y − 2x 9
<
+
3
3
3
− 2x
y<
+3
2
In the example, the slope is − and the y-intercept is 3.
3
3
Step 2: Determine if it is solid or dotted and graph the line. > or <
is dotted (doesn’t include the line) and > or < is solid (does
include the line).
Because it is < (less than), this
line will be dotted.
Step 3: Put your pencil on the y intercept. If it is greater than,
then shade greater y-intercept values (above the line). If it is less
than, then shade less y-intercept values (below the line).
© Copyright 2006 – LeAnn Barr
Shade below this line.
Example 2: 3 x − 2 y ≤ 4
Step 1: Solve for y. For this one, change the inequality symbol
when you divide by -2.
Step 2: Because it is >, this will be a solid line.
Step 3: Greater values are above the line.
Algebra I - TEXAS Style
52
3x − 2 y ≤ 4
− 3x
− 3x
− 2 y ≤ −3 x + 4
− 2 y − 3x 4
≤
+
−2
−2 −2
3
y ≥ x−2
2
Inequalities
Lesson 4-2
Try these:
1. 3 x + 2 y < 8
2. 2 x − 4 y ≥ 4
© Copyright 2006 – LeAnn Barr
4. Which of the following correctly represents this inequality?
5. Which of the following inequalities best represents
this graph?
1
x+2
2
1
B. y > − x + 2
2
1
C. y < x + 2
2
1
D. y < − x + 2
2
A. y >
Algebra I - TEXAS Style
53
3. x + 2 y ≥ 6
2x − y > 4
Inequalities – Two Variables
Lesson 4-3
© Copyright 2006 – LeAnn Barr
Estimate the height of the items in the following pictures. Record these at the top of a piece of
graph paper.
The smiling flower in each picture is actually 51.1 cm tall. Use the ruler on your
formula chart to find the actual height of each item.
Example: The dog: The flower in the picture is 7.7
cm tall and the dog is 4.6 cm tall. Put these values
in a proportion along with the actual measurement
of the flower and solve.
Algebra I - TEXAS Style
54
7 .7
4 .6
=
51 . 1
x
7 . 7 x = 235 . 06
7 .7
7 .7
x ≈ 30 . 5 cm
Inequalities – Two Variables
Lesson 4-3
Graph the estimated height and actual height of each item, (estimate, actual)
on your graph paper. Your graph should be labeled with the x axis (0 to
1000) with a scale of 50, and the y axis (0 to 1000) with a scale of 50. The x
axis represents your estimate, and the y axis represents the actual height of
the item
Now draw the line y=x. This is the line that represents a perfect estimate.
Did you underestimate or overestimate?
Look at the individual points to decide. Consider the point (200,100), where the estimate was
200 and the actual was 100. This point is below the line, but it is an overestimate.
Try these:
© Copyright 2006 – LeAnn Barr
Just Peachy Orchards set goals for the number of
bushels of peaches collected each week during the
harvest, and compared these to the actual number of
bushels harvested. Look at the graph at the right and
answer each question.
1. Which day(s) did they harvest more than the
goal?
2. Which day(s) did they harvest less than the goal?
3. Which day did they harvest exactly the same as
the goal?
Zoo tickets are $3 for adults and $1 for children. The Morgan family will spend $10.
4. Write an equation using x for adults and y for children to
represent spending exactly $10 on tickets.
5. Graph this line on a graph like the one at the right.
6. Which side of the line represents spending less than $10?
7. Which side of the line represents spending more than $10?
Algebra I - TEXAS Style
55
TAKS Practice
1. What will be the coordinates of ∆
A’B’C’, if ∆ ABC is reflected
across the y-axis?
A.
B.
C.
D.
(-4, -4), (-1, -2), (-3, -1)
(4, 4), (1, 2), (3, 1)
(-4, -4), (-2, -1), (-1, -3)
(4, 4), (2, 1), (1, 3)
4. The temperature on a January
night was -18º F, and began
increasing at 3ºper hour. Which
equation best represents this
situation?
A. T(x) = 3(x – 18)
B. T(x) = -18x + 3
C. T(x) = 3x – 18
D. T(x) = -18(x+3)
5. More Minutes phone company
charges a base rate of $20 and an
additional $0.05 per minute.
Cheaper Calls charges a base rate
of $15 and an additional $0.10
per minute. At how many
minutes do the two companies
charge the same amount?
© Copyright 2006 – LeAnn Barr
2. Country Views Cable company
charges $25 for basic channels
and $2.99 for each movie
channel. Which of the following
best represents this formula?
A.
B.
C.
D.
C(x) = 25x + 2.99
C(x) = 2.99x +25
C(x) = 25(x + 2.99)
C(x) = 2.99(x + 25)
3. A rectangle has an area of 100
square feet. If both dimensions
are divided by two, what is the
new area?
A.
B.
C.
D.
A.
B.
C.
D.
25
10
100
16
6. What is the solution to this
equation?
25 square feet
50 square feet
200 square feet
400 square feet
5x – 3 = 11 – 2x
A.
B.
C.
D.
Algebra I - TEXAS Style
56
2
-2
3
-9
Systems
5-1 Systems and Perimeter ………………………..page 58
a.
b.
c.
d.
Solving Perimeter Problems
Drawing and labeling a diagram
Set 5-1
TEKS – a1-6, b1C, b1D, b3A, b3B, b4A, b4B, c3A, c3B, c4A
New Numbered TEKS a1-6, A.1C, A.1D, A.3A, A.3B, A.4A, A.4B, A.7A, A.7B,
A.8A
5-2 Systems Finding Solutions …………………...page 60
a.
b.
c.
d.
Solving Systems using a Table
Solving Systems by Elimination
Set 5-2A, Set 5-2B
TEKS – Same as above plus c4B, c4C
New Numbered TEKS A.8B, A.8C
5-3 Systems and Graphing ……………………….. page 62
a. Solving for y and graphing on the calculator
b. Set 5-3A, Set 5-3B, Set 5-3C
c. TEKS – Same as above
© Copyright 2006 – LeAnn Barr
TAKS Practice ………………………………………... page 64
Algebra I - TEXAS Style
57
Systems
Perimeter Problems
Lesson 5-1
Example 1: The length of a rectangle is 4 cm more than the width. The perimeter is 20 cm. Find
the dimensions.
The first thing you do is draw the
shape. So draw a rectangle.
w
Now you label the sides. Since the
length is compared to width, use w for
the width, and w+4 for the length.
w+4
w+4
w
Add all the way around the rectangle. Put your answer equal to
the perimeter, 20, then solve the equation.
Sometimes you may need to write
the system of equations.
We use the formula for
perimeter, 2l+2w=P, and the
equation comparing the length
to the width.
4w+8 = 20
-8 -8
4w = 12
4 4
w=3
w=3
l=w+4
=3+4
=7
This one would be
© Copyright 2006 – LeAnn Barr
2l+2w=20
l=w+4
Example 2: The length of a rectangle is 3 more than twice the width. If the perimeter is 42 inches,
what are the dimensions of this rectangle?
Draw and label the rectangle.
2w+3
w
w
2w+3
Write an equation by adding around the rectangle, then solve.
6w+6 = 42
-6 -6
6w = 36
6 6
w=6
w=6
l = 2w + 3
= 2(6) + 3
= 15
Algebra I - TEXAS Style
58
Write the system of
equations:
2l + 2w = 42
l = 2w + 3
Systems
Perimeter Problems
Lesson 5-1
Try these:
1. The length of a rectangle is 8 in. more than the width. If the perimeter is 36 in., what are the
dimensions?
a. Draw and label the rectangle.
b. Write the equation by adding around the rectangle.
c. What are the dimensions?
d. Write the system of equations.
2. The length of a rectangle is 6 more than four times the width. If the perimeter is 82 cm, what are
the length and width?
a. Draw and label the rectangle.
b. Write the equation by adding around the rectangle.
c. What are the dimensions?
d. Write the system of equations.
3. A garden is fenced in using 5 wire mesh panels for the length and 3 wire mesh panels for the
width. The perimeter is 128 feet. Answer each question.
a. Draw and label a rectangle, use x for the length of one panel. (hint: 5x for the total length)
b. Write the equation by adding around the rectangle.
c. What is the length of one panel?
d. What is the total length and width of the rectangle?
e. Write the system of equations.
f. What is the area of the garden? (hint: use your dimensions from question d.)
TAKS Practice:
4. The length of a rectangle is 4 more than twice the width and the perimeter is 68 ft. Which
system of equations could be used to find the dimensions of this rectangle?
© Copyright 2006 – LeAnn Barr
A.
C.
l + w = 68
l = 4w + 2
2l + 2w = 68
l = 2w + 4
B.
D.
l + w = 68
l = 2w + 4
2l + 2 w = 68
w = 2l + 4
5. To build a flowerbed Carlos used 4 pieces of decorative edging along the front and back of the
flowerbed and 2 pieces of decorative edging plus 1 foot (he had to cut a piece) to create the sides.
He used a total of 38 feet of edging. Which of the following shows the correct length of a piece of
edging and the dimensions of this flowerbed?
A.
B.
C.
D.
edging = 3 ft, length = 12 ft, width = 7 ft
edging = 4 ft, length = 16 ft, width = 9 ft
edging = 5 ft, length = 20 ft, width = 11 ft
edging = 6 ft, length = 24 ft, width = 13 ft
Algebra I - TEXAS Style
59
Systems
Using a Table and Elimination
Lesson 5-2
Using a Table
1. A Little League Baseball team went to a
Ranger game.
Use the table to investigate
possible combinations of people
that satisfy the conditions.
*15 people went to the game.
*They spent exactly $132 for admission price.
*The tickets were $12.00 per adult and $8 per child.
Use the table to investigate possible combinations of people
that satisfy the conditions.
Number of
Adults
Number of
Children
1
14
Total
number of
people
15
Cost for
Adults
Cost for
Children
Total
Cost
12(1)
8(14)
124
Stop when
you find
it.
© Copyright 2006 – LeAnn Barr
Tables are not the only way to solve a
system of equations.
Look at the algebra tiles
that represent this system.
x + y = 10
2x − y = 8
Do you notice anything
that will zero out?
Can you cancel it out
without drawing the tiles?
x
Legend:
y
1
shaded = negative
Algebra I - TEXAS Style
60
Systems
Using a Table and Elimination
Lesson 5-2
Try these: Solve each of the following systems:
1.
x− y =3
x+ y =5
2.
3.
x + 3y = 7
3x + 3 y = 9
4.
5.
x + 3y = 7
x + 3 y = −4
6.
x + y = 13
x− y =7
5 x + 2 y = −8
3 x − 2 y = −8
4 x − y = −14
5 and 6 may require
some extra work.
After trying to solve
by elimination, solve
each for y and graph
on your calculator.
What do you notice?
8 x − 2 y = −28
© Copyright 2006 – LeAnn Barr
7. Write the system of equations that represents these tiles, then solve:
TAKS Practice:
8. Mrs. R. E. Derr has been keeping a tally of the number and types of books her students
have read so far this year. The types listed are mysteries, romance, technical, and
historical. She noticed that 55% were mysteries, 35% were romance, and 2 ½ % were
technical and 9 books were historical. How many mysteries were read so far this year?
A.
B.
C.
D.
120
75
66
7.5
9. Mr. B. R. Oddway is taking his family to the theater. Adult tickets cost $5 and children’s
tickets cost $3. He spent $27 on 7 tickets. Use the system of equations below to find how
many adults tickets he bought.
5 x + 3 y = 27
x+ y =7
A. 3
B. 4
C. 5
D. 6
Algebra I - TEXAS Style
61
Systems
Graphing
Lesson 5-3
Warm-up:
The sum of two numbers is 38 and their difference is 16. What are the two numbers?
a. Write the equation to represent the sum of these two numbers.
b. Write the equation to represent the difference of these two numbers.
c. What are the two numbers?
Example: A delivery truck is carrying exterior metal doors. Some of the doors have glass
and weigh 100 pounds and with packing take up 12 cubic feet of space. The rest of the
doors are insulated metal with no glass, weighing 75 pounds and taking up 6.75 cubic feet
of space. The total weight of the load is 3000 pounds and 315 cubic feet of space.
In this
problem,
nothing
easily
cancels
out.
a. Write an equation to represent the total number of
pounds. Use g to represent the doors with glass
and n to represent the doors with no glass.
b. Write an equation to represent the total cubic
space, using the same variables.
Lets change the variables
to x and y, then solve each for
y to use the calculator.
© Copyright 2006 – LeAnn Barr
Step 1: Solve each equation for y.
Step 2: Enter the first equation in
y1 and the second equation in y 2 .
y1 = −4 x / 3 + 40
y 2 = −16 x / 9 + 140 / 3
Step 3: Set your WINDOW to
[-100,100] by [-100,100] and GRAPH
100g + 75n = 3000
12g + 6.75n = 315
100x + 75y = 3000
-100x
-100x
75y = -100x + 3000
75
75
75
y = -4x + 40
3
12x + 6.75y = 315
-12x
-12x
6.75y = -12x + 315
6.75
6.75 6.75
y = -16x + 140
9
3
Step 4: 2nd,TRACE, choose 5:
intersect.
Step 5: First curve? ENTER. Second
curve? ENTER. Guess? ENTER.
Step 6: Write down your answer.
Intersection
X=15
y=20
c. The truck carried 15 doors with glass and 20 doors without glass.
Algebra I - TEXAS Style
62
Systems
Graphing
Lesson 5-3
Try these:
1.
5 x + 3 y = 5.75
2.
4 x − 2 y = −2
2 x + 7 y = −8
5 x − 3 y = 21
3. The owners of The Little Fruit Stand made a new sign for their prices. However,
before the paint dried, a rainstorm washed all the prices away. They had these
partial receipts that used the new prices.
#2021
1 blueberry box
1 strawberry box
total
$4.00
#2022
2 blueberry boxes
1 strawberry box
total
$6.25
#2023
2 strawberry boxes
2 peach boxes
total
$6.00
a. Write the equation to represent each receipt.
#2021
#2022
#2023
b. What is the cost per box of each?
Blueberries
Strawberries
Peaches
© Copyright 2006 – LeAnn Barr
4. Bill is making a tile design. The first day he bought 5 red tiles and 3 blue tiles and spent
$53. As most artwork goes, Bill discovered he needed 4 more red and 5 more blue, and
spent $58. Over the weekend, Bill almost finished but was short 1 red and 1 blue tile.
a. Write the equation for the first day’s expense on tiles.
b. Write the equation for the second day’s expense on tiles.
c. How much did a red tile cost?
d. How much did a blue tile cost?
e. How much more will Bill have to spend to finish his design?
5. Bill has 23 coins, all dimes and quarters, worth $3.05.
a. Write an equation to represent the total number of coins.
b. Write an equation to represent the value of the coins.
c. How many of each type of coin does he have?
Algebra I - TEXAS Style
63
TAKS Practice
1. The perimeter of a rectangular garden is
60 ft, and the length is 10 ft less than
twice the width. Which system of
equations could be used to find the
dimensions of this garden?
5. Carpet installation is figured using the
following formula
C ( x ) = 0.50 x + 75 ,
where x represents the number of
square feet. What is the independent
variable?
l + w = 60
A.
w = 2l − 10
B.
C.
D.
A.
B.
C.
D.
l + w = 60
l = 10 − 2 w
2l + 2w = 60
w = 2l − 10
6. Which of the following methods would
best be used to find the solution of this
equation by graphing?
2l + 2w = 60
l = 2w − 10
8 x + 3 = −2 x − 17
2. (x,3) is a solution for the equation
2x+3y=7, what is the value of x?
© Copyright 2006 – LeAnn Barr
A.
B.
C.
D.
A. Graph y=8x+3 and y=-2x-17
and find the y value of the
point of intersection.
B. Graph y=8x+3-2x-17 and find
the x-intercept.
C. Graph y=8x+3 and y=-2x-17
and find the x value of the
point of intersection.
D. Graph y=8x+3-2x-17 and find
the y-intercept.
-1
1
2
3
3. A square tile has an area of 36 in 2 .
How many tiles would be needed to
cover a square floor with a perimeter
of 144 inches ?
A.
B.
C.
D.
4
36
48
144
4. The cost of an advertisement in a
newspaper depends on the number of
characters purchased. An ad with 30
characters cost $19, while an ad with
25 characters cost $16.50. What is the
cost per character?
A.
B.
C.
D.
square feet
total cost
installation fee
$0.50
7. The surface area of a globe is
0.181 m². If a larger model of this
globe is created for a display by
multiplying the radius by 2, what is the
surface area of the larger model?
$2.50
$0.50
$4
$5
Algebra I - TEXAS Style
64
A.
B.
C.
D.
0.0905 m²
0.362 m²
0.724 m²
1.448 m²
Exponents
6-1 Exponents ……………………………………………… page 66
a. Discovering Rules for Exponents by Expanding.
b. Workbook Set 6-1A (Bright Lights), Set 6-1B
c. TEKS a1-6, b2C, b3A, b3B, d3A, d3C
New Numbered TEKS a1-6, A.2C, A.3A, A.3B, A.11A, A.11C
6-2 Exponents and Division …………………………… page 68
a.
b.
c.
d.
e.
Division and exponents
Scientific Notation
Multiplying and Dividing with Scientific Notation
Workbook Set 6-2A (Paper Folding), Set 6-2B, Set 6-2C
TEKS same as above
6-3 Negative Exponents ………………………………… page 70
a. Negative Exponents
b. Workbook Set 6-3
c. TEKS same as above
© Copyright 2006 – LeAnn Barr
TAKS Practice …………………………………………page 72
Algebra I - TEXAS Style
65
Exponents
Lesson 6-1
We’re going to expand 32 in the box at
the right, then simplify.
32 = 3 ⋅ 3
=9
The 3 is the base and the 2 is
the exponent.
So the exponent tells you
to multiply the base times
itself that many times.
Try these:
1
2. ⎛⎜ ⎞⎟
⎝ 4⎠
1. (2)
3
2
2
3. ⎛⎜ ⎞⎟
⎝5⎠
3
It works the same way with
variables. Look at this one.
n3 ⋅ n = (n ⋅ n ⋅ n) ⋅ (n)
= n4
Expand, then write the
base with the new
exponent.
Try these, expand each, then simplify:
© Copyright 2006 – LeAnn Barr
4. (x 2 )(x 4 )
(a 3 ) 2 = a 3 ⋅ a 3
= ( a ⋅ a ⋅ a )( a ⋅ a ⋅ a )
= a6
Try these, expand each, then simplify:
7. (x 3 ) 4
6. (c 2 d 3 )(cd 2 )
5. w ⋅ w5
Now look what
happens when an
exponent is raised
to a power.
Just expand it,
then expand
again.
8. (m 2 ) 3
Algebra I - TEXAS Style
66
9. (4 y 3 ) 2
Expand these
separately:
(ccddd)(cdd)
Exponents
Lesson 6-1
Using the patterns you found on the previous page, copy and complete these
conjectures.
• When multiplying two of the same variables with exponents, write down the
base and ____________ (add, subtract, multiply, or divide) the exponents to
find the new exponent.
m
n
m ? n
x ⋅x = x
What operation goes
here? +, -, • , ÷ ?
• When raising an exponent to a power, write down the base and
____________(add, subtract, multiply, or divide) the exponents to find the
new exponent.
m n
m ? n
© Copyright 2006 – LeAnn Barr
(x ) = x
What operation goes
here? +, -, • , ÷ ?
Try this one:
10. Marine Biology students took a sample of a plant and placed it in a tank. Each
week they measured the amount of surface this plant covered in the tank and
recorded it in the table below.
Week
Process
Area
a. Identify the independent and
in cm 2
dependent variables.
0
2.5
1
2.5 ⋅ 2 = 2.5 ⋅ 2
1
5
2
b. Describe the relationship between
2.5 ⋅ 2 ⋅ 2 = 2.5 ⋅ 2
2
10
3
the variables.
3
2.5 ⋅ 2 ⋅ 2 ⋅ 2 = 2.5 ⋅ 2
20
c. If this process continues, how much area will be covered by week 4?
d. What week would you expect to cover 160 cm 2 ?
e. What is the function rule? Explain.
f. Is this linear? Explain.
g. How would the function rule be different if they had started with 4 cm 2 ?
h. How would the function rule be different if they noticed it tripled each week?
Algebra I - TEXAS Style
67
Exponents
Example 1:
Lesson 6-2
1
1
Expand the fraction, on top
and on the bottom.
1
5
w
w⋅ w⋅ w⋅ w⋅ w
=
w⋅ w⋅ w
w3
1 1 1
w2
=
1
= w2
Example 2:
3
Cross out the ones that
divide and equal 1.
Write the ones that are
left in exponential form.
Example 3:
1 1 1
3
© Copyright 2006 – LeAnn Barr
w
w⋅ w⋅ w
=
w5 w ⋅ w ⋅ w ⋅ w ⋅ w
1 1 1
1
= 2
w
1
1
1
w w⋅ w⋅ w
=
3
w w⋅ w⋅ w
1 1 1
=1
= w3−3 = w0 = 1
Use the patterns from the examples to copy and complete the rules.
• When dividing with exponents that have the same base, write the base and ___________ (add,
subtract, multiply, or divide) to find the new exponent.
• If the larger exponent is on _________ (top or bottom), then write the base, with the new
exponent. (See example 1)
• If the larger exponent is on _________(top or bottom), then make a fraction with one as
the numerator and the base and exponent as the denominator. (See example 2)
(The base and exponent go where the biggest exponent is)
•
•
When dividing two exponents with the same base and the exponent is the same the value
is ______________. (See example 3)
Any number raised to the 0 power is equal to ___________. (See example 3)
Try these, expand and simplify.
25
22
c5
4. 2
c
1.
c3d
7. 4 4
c d
3
33
n2
5. 6
n
2.
4x3 y 6
8.
8x3 y 5
When there isn’t an
exponent written,
the exponent is 1.
Simplify the fraction
4
like always, then
8
simplify the rest.
Algebra I - TEXAS Style
68
56
56
y4
6. 4
y
3.
9.
10w 3 x 2
2x
Exponents
Lesson 6-2
Now lets write some numbers in scientific notation.
Scientific notation has a number that
is between 1 and 10 times 10 to a
power. Look at these examples.
Example 4: Write 5,230 in scientific notation.
5,230 = 5.23 × 1000
= 5.23 × 10 3
3
Notice that 1000 = 10 , and the decimal
moved 3 places to the left.
Example 6: Write 8.75 × 10 4 in standard
notation.
8.75 × 10 4 = 8.75 × 10000
= 87,500
© Copyright 2006 – LeAnn Barr
Notice that the decimal moved 4 places
to the right (to make a number >1).
Example 5: Write 0.0471 in scientific notation.
0.0471 = 4.71× 0.01
1
= 4.71 ×
100
1
= 4.71 × 2
10
= 4.71× 10 − 2
1
Notice that 0.01 =
= 10 −2 and the
2
10
decimal moved 2 places to the right. We
will learn more about negative exponents
in the next lesson.
Example 7: Write 9.21× 10 −3 in standard
notation.
1
9.21× 10 −3 = 9.21×
1000
= 9.21× 0.001
= 0.00921
Notice that the decimal moved 3 places
to the left (to make a number <1).
Try these:
10. Write 3,840,000 in scientific notation.
11. Write 0.00000409 in scientific notation.
12. Write 9.12 × 10 −5 in standard notation.
13. Write 5.701× 10 3 in standard notation.
14. The area of a rectangle is 36 x 5 y 4 and the length is 9 x 3 y . What is the width?
Algebra I - TEXAS Style
69
Exponents
Lesson 6-3
53
by subtracting
55
the exponents.
53
Simplify 5 by
5
expanding.
Simplify
Now compare your answers
in the calculator.
1
5 −2 =
25
Try these:
1
2. ⎛⎜ ⎞⎟
⎝4⎠
1. (3)
−2
−2
3
3. ⎛⎜ ⎞⎟
⎝4⎠
−3
Lets use the same
properties with variables.
Example 3:
5m −2 = 5(m −2 )
Example 2: m −2 =
Example 5:
(3m )
© Copyright 2006 – LeAnn Barr
3 −2
=
=
1
(3m )
3 2
1
m2
Example 4:
1
(3m )2
1
=
9m 2
(3m )−2 =
⎛ 1 ⎞
= 5⎜ 2 ⎟
⎝m ⎠
5
= 2
m
Example 6:
4c 3
3 −2
4c d = 2
d
Example 7:
6x 2 y 5 3 y 4
=
8x 5 y 4 x 3
1
9m 6
Try these, use only positive exponents in your answer:
4. (4 x 4 )(3 x 3 )
5. (3m 2 )
6. v −4
7. (4c )−2
8. 4c −2
9.
(3c )
10. w −3 x 7
11. 6k 2 m −4
12.
9a 3 b
6a 2 b 3
3
Algebra I - TEXAS Style
70
4 −2
Exponents
Lesson 6-3
TAKS practice:
13. One strand of a cable has a cross
section area of 1.2 × 10 −1 cm 2 . If it
takes 100 strands to make one
cable, what is the total area of a
cross section of this cable?
Remember that
A.
B.
C.
D.
100 = 10
14. Evaluate the following expression if
m = -3:
m5
m9
2
1.2 × 10 −3 cm 2
1.2 ×10 −1 cm 2
1.2 ×10 0 cm 2
1.2 ×101 cm 2
1
81
1
B.
− 81
1
C.
12
1
D.
− 12
A.
15. Sara invested $1000 in an account that paid 1% each month. She recorded her earnings
in a table.
Month
0
1
2
3
Total value
1000
1010
1020.10
1030.30
Which of the following equations represents this problem?
© Copyright 2006 – LeAnn Barr
A.
B.
C.
D.
V
V
V
V
= (1.01) n 1000
= (0.01) n 1000
= (0.99) n 1000
= (1000 + 0.01) n
16. On what month will the value be more than 120% of the starting value?
A.
B.
C.
D.
8th
15th
19th
29th
17. A triangle has its boundary along the x-axis and y-axis and the line 5x-2y=10. What is the
area of the triangle?
A.
B.
C.
D.
5
10
15
20
Algebra I - TEXAS Style
71
TAKS Practice
5. The area of a garden is 40 sq ft.
The owner wants to enlarge the
garden by doubling both
dimensions. What is the new
area of the garden?
A. 10
B. 20
C. 80
D. 160
1. Evaluate the following
expression if a = -4:
a3
a2
1
4
B. − 4
1
C.
4
D. 4
A. −
6. Which of the following equations
best represents this table?
x
y
2. What is the solution to the
system of equations
3x + 2 y = 6
5 x − 2 y = −6
A.
B.
C.
D.
A.
B.
C.
D.
(3,0)
(0,3)
(3,2)
(8,0)
© Copyright 2006 – LeAnn Barr
3. What is the y-intercept of the
equation y = 2(3 x − 1) ?
A.
B.
C.
D.
2
8
3
11
4
14
5
17
y = 3x + 1
y = 3x + 2
y = x+5
y = 3x − 2
7. Jill wants to create a new
equation from this table by
doubling the slope and
decreasing the y-intercept by 1.
x
y
-1
-2
6
5
1
5
2
8
3
11
4
14
5
17
What is the Jill’s new equation?
A. y = 6 x + 1
3
B. y = x + 1
2
C. y = 2 x + 4
D. y = 3 x − 2
4. What is the slope of the
equation y = 2(3 x − 1) ?
A.
B.
C.
D.
1
5
-1
-2
6
5
Algebra I - TEXAS Style
72
Polynomials
7-1 Polynomials – Simplifying …………………….page 74
a.
b.
c.
d.
Adding and Subtracting Polynomials
With and Without Algebra Tiles
Workbook Set 7-1A and 7-1B
TEKS a1-6, b4A, b4B
New Numbered TEKS – a1-6, A.4A, A.4B
7-2
Polynomials using the Distributive Property
………………….page 76
a.
b.
c.
d.
7-3
Distributive Property
With and Without Algebra Tiles
Workbook Set 7-2A and 7-2B
TEKS same as above
Multiplying Polynomials …………………….page 78
a. With and without Algebra Tiles
b. Workbook Set 7-3A, 7-3B, 7-3C (parametric changes
activity), and 7-3D (review)
c. TEKS same as above for 7-3A, 7-3B and 7-3D
d. Set 7-3C TEKS a1-6, b2A, d1B, d1C, d1D
New Numbered TEKS – a1-6, A.2A, A.9B, A.9C, A.9D
© Copyright 2006 – LeAnn Barr
TAKS Practice ………………………………………..page 80
Algebra I - TEXAS Style
73
Polynomials
Add and Subtract
This is a legend of Algebra Tiles,
copy this in your notes.
Lesson 7-1
Legend
x2
x
1
-x2
-x
-1
Example 1: Draw the Algebra Tiles to represent this problem and simplify by
combining the ones that are the same size and shape.
Then work the same problem by stacking the like terms. Compare your answers.
( x 2 − 2 x + 3) + (2 x 2 − x − 5)
Notice when combining the x2 s, you
get 2x2.
When combining the xs, you get -3x.
2 x 2 − 3x − 2
And when combining the 1s, some
cancel out, which leaves you with -2
This is the same
problem worked by
stacking it up. Line
up the like terms
and then add.
x2 − 2x + 3
+ x2 − x − 5
2 x 2 − 3x − 2
© Copyright 2006 – LeAnn Barr
1. Try this one: (3 x 2 + x − 2) + (2 x − 3)
Example 2: Draw the Algebra Tiles to represent this problem, remember to
distribute the – before you draw the tiles, and simplify by combining the ones that
are the same size and shape.
Then work the same problem by stacking the like terms. Compare your answers.
( x 2 − 2 x + 1) − (2 x 2 − x + 4)
Before you draw the tiles for this one
you must distribute a -1 through the
second set of parentheses. (Changing
all the signs in that set of parentheses.)
Then just follow the same steps as
above.
− x2 − x − 3
2. Try this one: ( 2 x 2 + 2 x − 3) − (3x + 5)
Algebra I - TEXAS Style
74
This is the same
problem worked by
stacking it up. Line
up the like terms
and then add.
x2 − 2x +1
− 2x2 + x − 4
− x2 − x − 3
Try these without using tiles:
3. (3x 2 + 2 x − 5) + (2 x 2 − 4 x − 3)
4. (3 x 2 + 2 x − 5) − (2 x 2 − 4 x − 3)
5. (4 x 2 − 7) + (3 x + 2)
6. (4 x 2 − 7) − (3x + 2)
7. (12 x 2 + 3 x − 11) + (3 x 2 − 4)
8. (12 x 2 + 3 x − 11) − (3x 2 − 4)
9. Find each value using f ( x) = 2 x 2 − 3 x − 1 and g ( x) = x 2 − 3
a. f (1)
b. f (− 2)
c. f (0)
d. g (3)
e. g (−4)
f. g (0)
g. f ( 4) + g (2)
h. f ( 4) − g (2)
© Copyright 2006 – LeAnn Barr
10. The lengths of the sides of a pentagon are 2x-1, 3x+1, x+1, x-3, and 2x.
Write the algebraic expression for the perimeter, P.
11. The perimeter of a triangle is 8c-1. Two of the sides are represented by the
expressions 2c+3 and 4c-1. What is the length of the third side?
12. Profit is figured by subtracting Cost (C), from Revenue (R). Unique
Expressions prints t-shirts. They figured their revenue to be R = x 2 + 15 x + 10 ,
and the cost to be C = x 2 + 10 x + 50 for x number of t-shirts. What is the
expression for profit?
Algebra I - TEXAS Style
75
Polynomials
Distributive Property
Lesson 7-2
We are going to begin by sketching the
Algebra Tiles that represent each part
of the problem, then fill in the area with
tiles that fit.
Example 1: 2(x+1)
Step 1: Place the tiles that represent
the length and width of the rectangle
on each side of the rectangle.
Step 2: Draw imaginary lines at the
edges of each tile.
Step 3: Place the appropriate tiles in
the boxes.
Step 4: Your answer is 3x+3
Example 2: Now lets work the same problem without tiles:
3( x + 1) = 3 ⋅ x + 3 ⋅1
© Copyright 2006 – LeAnn Barr
= 3x + 3
Example 3: Work the following using tiles: x(x-1)
x2 − x
Example 4: Work example 3 without tiles.
x ( x − 1) = x ⋅ x − x ⋅1
= x2 − x
Algebra I - TEXAS Style
76
Try these by drawing tiles:
1. 2(x-1)
2. x(x+2)
3. 2x(x-2)
Try these without using tiles:
4. 4(2x-5)
5. 2x(3x+2)
6. -5(2x-1)
7. -3x(x+2)
8. 3(5x-1)
9.
1
(4 x − 6)
2
Find the dimensions of each rectangle pictured below, then find its area.
10.
11.
12.
Write the problem pictured in tiles below and find the answer.
© Copyright 2006 – LeAnn Barr
13.
14.
15. The length of a rectangle is 3 more than the width.
a. Draw and label a diagram.
b. Find the expression for the area, then simplify. A=
c. Find the expression for the perimeter, then simplify. P=
Algebra I - TEXAS Style
77
Polynomials Multiplying
Lesson 7-3
Now we are going to multiply
polynomials. Start by sketching the
Algebra Tiles that represent this
problem, then write the answer.
Example 1:
(x+1)(x+3)
Area = x 2 + 3 x + 1x + 3
= x2 + 4x + 3
Try these: Sketch the Algebra Tiles that represent each problem, then write the
answer.
1. (x+2)(x+3)
2. (x-3)(x+1)
3. (x-1)(x-2)
4. (2x-1)(x+2)
© Copyright 2006 – LeAnn Barr
12
To Multiply without using tiles, try
stacking them up like multiplying a two
digit number by a two digit number.
× 14
48
12
168
Example 2: (x+3)(x+4)
x+3
x+4
4 x + 12
x 2 + 3x
x 2 + 7 x + 12
Algebra I - TEXAS Style
78
Multiply 4 times (x+3).
Then multiply x times (x+3).
Add your like terms.
Try these without using tiles.
5. (x-4)(x+5)
6. (x+6)(x-3)
7. (2x-1)(x-3)
8. (5x+2)(2x+5)
9. The length of the flowerbed took 2 feet more than 3
sections of edging. The width required cutting two feet
off of a section of edging.
a. Label using x for the length of one section of
edging.
b. Write the expression for area. A=
c. Write the expression for perimeter. P=
d. The perimeter was 40 ft. Find the length and width.
e. Find the area.
© Copyright 2006 – LeAnn Barr
10. Find the surface area of the rectangular prism.
Remember that surface area is the sum
of the areas of each face.
Front =
Back =
Top =
Bottom =
Left Side =
Right Side = ___________
11. Find the volume of the rectangular prism.
Volume of a Rectangular Prism = lwh
Algebra I - TEXAS Style
79
For questions 10 and 11.
x
x+2
x+1
TAKS Practice
1. The length of the rectangle shown below is (2x+1) and its width is (3x-1). What
is the area of this rectangle?
Legend
x2
-x2
A.
B.
C.
D.
x
-x
1
-1
6 x 2 + 5x + 1
6x 2 + x − 1
6x 2 + x + 1
5x
© Copyright 2006 – LeAnn Barr
2. Find the perimeter of the rectangle shown above.
A. 6 x 2 + 10 x + 1
B. 12 x 2 + 2 x − 2
C. 5 x − 2
D. 10 x
3. Find the value of f (−2) in the function f ( x) = 3x 2 − x − 4 .
A.
B.
C.
D.
10
6
-18
-16
4. The perimeter of a room on a blueprint is 6 inches. If the scale is ¼ in = 1 ft, what
is the perimeter of the actual room?
A.
B.
C.
D.
10 ft
24 ft
1.5 ft
12 ft
Algebra I - TEXAS Style
80
Solving Quadratics
8-1 Exploring Factoring …………………………page 82
a.
b.
c.
d.
Finding a common factor, roots, and y-intercepts.
Using Algebra Tiles and Graphing.
Workbook Set 8-1
TEKS b4A, b4B, d2A, d2B
New Numbered TEKS A.4A, A.4B, A.10A, A.10B
2
8-2 Factoring x ± bx + c ………………………...page 84
a. Workbook Set 8-2A and 8-2B.
b. TEKS Same as above.
2
8-3 Factoring x ± bx − c …………………………page 86
a. Workbook Set 8-3A and 8-3B.
b. TEKS Same as above.
8-4 Factoring Squares and Difference of Squares
…………….page 88
a. Factoring Squares and Differences of Squares
b. Workbook Set 8-4
c. TEKS Same as above
© Copyright 2006 – LeAnn Barr
2
8-5 Factoring ax ± bx ± c ………………………..page 90
a. Workbook Set 8-5
b. TEKS Same as above.
8-6 Solving Quadratic Equations ………………page 92
a.
b.
c.
d.
Quadratic Formula
Verifying by Graphing
Workbook Set 8-6A and 8-6B.
TEKS Same as above
Algebra I - TEXAS Style
81
Factoring
Lesson 8-1
Example 1: Find the missing terms in each of the following problems.
a. 5 x 2 − 10 x = 5 x( ? )
5 x 2 10 x
−
= x−2
Solution:
5x 5x
Check: 5 x( x − 2) = 5 x 2 − 10 x
b. x 2 + 12 x + 20 = ( x + 2)( ? )
Solution: Think, x times what is x2? x, so (x + ?)
Now, 2 times what number is 20? 10, so (x + 10).
x+2
x + 10
Check:
10 x + 20
2
x + 2x
x 2 + 12 x + 20
c. 6 x 2 − 10 x − 4 = (3 x + 1)( ? )
Solution: Think 3x times what is 6x2. (2x+?)
Now, 1 times what number is - 4? (2x-4)
3x + 1
2x − 4
© Copyright 2006 – LeAnn Barr
Check:
− 12 x − 4
6x2 + 2x
Try these: Be sure to check your answers.
6 x 2 − 10 x − 4
1. 12 x 2 + 20 x = 4 x ( ? )
2. 14 x 2 − 21x = ? (2 x − 3)
3. 5 x 2 + 25 x = ? ( x + 5)
4. 9 x 2 + 12 x = 3 x( ? )
5. x 2 − 4 x − 12 = ( x + 2)( ? )
6. x 2 + 5 x − 14 = ( x − 2)( ? )
7. 2 x 2 + x − 10 = ( x − 2)( ? )
8. 2 x 2 + 9 x + 10 = ( x + 2)( ? )
Algebra I - TEXAS Style
82
Factoring
Lesson 8-1
Example 2: Find the factors of x 2 + 2 x .
Using Algebra Tiles:
Graphing:
x+2
x
Roots are on the x-axis.
Factors: x (x+2)
Roots solutions and zeros are the same thing:
Take each of the factors and set them equal to 0, and solve:
x = 0 and
x+2=0
x = −2
So the roots are x = 0, x = -2. Compare these to the zeros on the graph
above.
© Copyright 2006 – LeAnn Barr
Try these, draw the Algebra Tiles, Graph, then identify the factors and the roots.
9. x 2 + 4 x
10. x 2 − 4 x
11. 2 x 2 + 6 x
(hint: 2x is a factor)
12. 3x 2 + 9 x
13. 8 x 2 − 4 x
14. 6 x 2 + 4 x
(sometimes the roots are fractions)
15. Use the graph to find the factors, area, and roots of the
equation.
Algebra I - TEXAS Style
83
Factoring
x 2 ± bx + c
Lesson 8-2
Example1: Draw the tiles to represent this multiplication problem. Then stack and
multiply.
(x-2)(x-3)
x−2
x−3
− 3x + 6
x2 − 2x
x2 − 5x + 6
Example 2: Now let’s take the polynomial and find the factors using Algebra Tiles.
x2 − 5x + 4
x-4
x-1
Checking:
This didn’t
work, there is
extra space.
This worked,
it used all the
pieces and
(-1)(-4)=4
x−4
x −1
−x+4
x2 − 4x
© Copyright 2006 – LeAnn Barr
x2 − 5x + 4
So the factors are (x-4)(x-1). Multiply to check.
Notice that (-1)(-4) = 4 (the last number) and -1 + -4 = -5 (the coefficient of x).
the
Example 3: Graph the function using your calculator and copy to graph paper. Remember
roots are on the
Label the x-intercepts, which are the same as solutions, roots and zeros.
x-axis.
Also label the y-intercept. How does this compare to the factors you found above?
You will notice the roots are
x = 1 and x = 4, when y = 0.
x −1 = 0
+1 +1
x−4=0
+4 +4
x =1
x=4
The y intercept is the value when x = 0,
0 2 − 5(0) + 4 = 4
Algebra I - TEXAS Style
84
Example 4: Use the patterns found above to find the factors and roots of y = x 2 + 7 x + 10.
1 ⋅10
2⋅5
The sets of factors for 10 are 1⋅10 and 2 ⋅ 5 , and 2 + 5 = 7
Therefore the factors must be (x+2)(x+5) and the roots are x = -2 and x = -5.
Check with Algebra Tiles, Draw a Graph with labels, and Multiply to check.
x+2
x+5
5 x + 10
2
x + 2x
x 2 + 7 x + 10
© Copyright 2006 – LeAnn Barr
Try these:
Find the factors, roots, and y-intercept of each. Verify your answer by drawing Algebra
Tiles, drawing a graph with labels, and multiplying to check.
1. y = x 2 + 8 x + 7
2. y = x 2 − 9 x + 8
3. y = x 2 − 8 x + 12
4. y = x 2 + 12 x + 20
5. Use the Algebra Tiles at the right to find the polynomial,
factors, roots, y-intercept, and graph with labels.
6. Use the graph at the left to find the polynomial, factors,
roots, y-intercept, and draw the Algebra Tiles.
7. The roots of a polynomial are x = -2 and x = - 6. Find the
factors, polynomial, y-intercept, draw a labeled graph and
draw the Algebra Tiles for this problem.
Algebra I - TEXAS Style
85
Factoring
x 2 ± bx − c
Lesson 8-3
Example1: Draw the tiles to represent this multiplication problem. Then stack and
multiply.
(x-2)(x+3)
Notice that the x’s
are opposites and
some cancel out
when simplifying.
3x – 2x = x
x−2
x+3
3x − 6
x2 − 2x
x2 + x − 6
Example 2: Now let’s take a polynomial and find its factors using Algebra Tiles.
x+4
x 2 + 3x − 4
Checking:
x+4
x-1
x −1
−x−4
x2 + 4x
x 2 + 3x − 4
These two pieces were added as
0, -x+x=0. This is the part that
will cancel out when simplifying.
© Copyright 2006 – LeAnn Barr
So the factors are (x+4)(x-1). Multiply to check.
Notice that (-1)(4) = 4 (the last number) and -1 + 4 = 3 (the coefficient of x).
the
Example 3: Graph the function using your calculator and copy to graph paper. Remember
roots are on the
Label the x-intercepts, which are the same as solutions, roots and zeros.
x-axis.
Also label the y-intercept. How does this compare to the factors you found above?
You will notice the roots are
x = 1 and x = -4, when y = 0.
x −1 = 0
+1 +1
x+4=0
−4 −4
x =1
x = −4
The y intercept is the value when x = 0,
0 2 + 3(0) − 4 = −4
Algebra I - TEXAS Style
86
Example 4: Use the patterns found above to find the factors and roots of y = x 2 − 9 x − 10.
1 ⋅10
2⋅5
Since 10 is negative we are looking for a difference of -9.
The sets of factors for 10 are 1⋅10 and 2 ⋅ 5 , and − 10 + 1 = −9
Therefore the factors must be (x+1)(x-10) and the roots are x = -1 and x = 10.
Check with Algebra Tiles, Draw a Graph with labels, and multiply to check.
x − 10
x +1
x − 10
2
x − 10 x
x 2 − 9 x − 10
© Copyright 2006 – LeAnn Barr
Try these:
Find the factors, roots, and y-intercept of each. Verify your answer by drawing Algebra
Tiles, drawing a graph with labels, and multiplying to check.
1. y = x 2 − 6 x − 7
2. y = x 2 + 2 x − 8
3. y = x 2 − 4 x − 12
4. y = x 2 + x − 20
5. Use the Algebra Tiles at the right to find the polynomial,
factors, roots, y-intercept, and graph with labels.
6. Use the graph at the left to find the polynomial, factors,
roots, y-intercept, and draw the Algebra Tiles.
7. The roots of a polynomial are x = -1 and x = 6. Find
the factors, polynomial, y-intercept, draw a labeled graph
and draw the Algebra Tiles for this problem.
Algebra I - TEXAS Style
87
Factoring
Difference of Square and Perfect Squares
Lesson 8-4
Example1: Factor x 2 − 6 x + 9 , draw Algebra Tiles, Graph, find the roots and y-intercept,
then check by multiplying.
Checking:
Notice that this is a
square, and its
factors are
( x − 3)( x − 3)
x −3
x −3
− 3x + 9
= ( x − 3) 2
2
x − 3x
x2 − 6x + 9
Factors: (x-3)(x-3) or (x-3)2
Root: x = 3, this graph only touches the x axis.
Y intercept: 9
Example 2: Factor x 2 − 9 , draw Algebra Tiles, Graph, find the roots and y-intercept,
then check by multiplying.
Checking:
This type of
problem is
known as a
difference of
squares.
x−3
x+3
3x − 9
© Copyright 2006 – LeAnn Barr
2
Factor the same
way as before:
x 2 + 0x − 9
Factors: (x-3)(x+3)
Roots: x = 3 and x = -3
Y-intercept: -9
Try these:
Factor each, graph, find the roots and y-intercept, then check by multiplying.
1. x 2 − 25
2. x 2 − 10 x + 25
3. x 2 − 36
4. x 2 + 12 x + 36
Algebra I - TEXAS Style
88
x − 3x
x2
−9
5. x 2 − x − 30
6. x 2 + 11x + 30
7. x 2 − 17 x + 30
8. x 2 + 7 x − 30
© Copyright 2006 – LeAnn Barr
Use each graph to identify the roots, factors and function rule. Identify the line of
symmetry.
9.
10.
11.
12.
13. The area of a rectangle is
x 2 + 9 x + 20 . What is the perimeter?
14. Find the larger solution of
x 2 + 8 x − 20 .
A. 4 x + 18
B. 9 x + 20
C. 2 x + 9
D. 4 x 2 + 36 x + 80
A.
B.
C.
D.
Algebra I - TEXAS Style
89
-10
10
2
-2
Factoring
ax 2 ± bx ± c
Lesson 8-5
Example 1: Factor 2 x 2 + 7 x + 3 = 0 . Find the solutions, draw the Algebra Tiles,
and graph with labels. Check by multiplying.
Multiply the coefficient of x 2 , 2 by the whole number, 3. Then list its factors.
1x
2x2
(?+ ?)(?+ ?)
( x + ?)(?+ 1)
6
1⋅ 6
2⋅3
1+6 = 7
6x
( x + ?)(2 x + 1)
6x
Factors: (x+3)(2x+1)
Roots (solutions): x = -3 and x = −
1
2
Checking:
2x +1
x+3
6x + 3
2
2 x + 1x
© Copyright 2006 – LeAnn Barr
2x2 + 7x + 3
Try these:
Factor, find the roots, draw the graph with labels, and check by multiplying.
1. 3x 2 − 13x − 10
2. 5 x 2 + 11x + 2
3. 4 x 2 + 15 x − 4
4. 2 x 2 − 11x − 6
Hint: If you have trouble, use your calculator and get an integer root from the
graph. Each of the above problems has a fraction and an integer as solutions.
Algebra I - TEXAS Style
90
Factoring
ax 2 ± bx ± c
Lesson 8-5
Try these:
Factor, find the roots, draw the graph with labels, and check by multiplying.
Each of these have fractions as solutions.
5. 8 x 2 + 26 x + 15
6. 6 x 2 − 1x − 2
2
1
and . Find the factors, the
3
2
polynomial, draw the graph with labels, and draw the Algebra Tiles.
Example 2: The roots of a polynomial are −
Solution: Work backward to find each factor.
2
3
2
3⋅ x = − ⋅3
3
3 x = −2
+2 +2
x=−
3x + 2 = 0
Multiply both sides of the equation by the
denominator, 3.
To remove -2, add 2 to both sides of the equation.
−1 −1
2x −1 = 0
One factor is (3x+2).
Repeat with the other solution.
© Copyright 2006 – LeAnn Barr
Factors : (3x+2) (2x-1)
Polynomial: 6 x 2 + x − 2
Try these: Find the factors and the polynomial for each problem.
7. Roots of x =
4
1
and x =
3
3
1
2
1
2⋅ x = ⋅2
2
2x = 1
x=
8. Zeros of
Algebra I - TEXAS Style
91
1
2
and −
3
5
Solving Quadratic Equations
Lesson 8-6
The quadratic formula is used for any quadratic
equation in the form of ax 2 + bx + c = 0 .
It is x =
− b + b 2 − 4ac
− b − b 2 − 4ac
and x =
2a
2a
a is the coefficient of x 2
b is the coefficient of x
and c is the other number.
Remember Coefficient means
the number in front of a
variable.
Step1. Identify a, b and c.
20 x 2 − 7 x − 6 = 0
a= 20, b=-7, c=-6
Step 2. Substitute into the formula.
x=
Step 4. Verify your answer by
graphing and finding the x-intercepts.
(− − 7 + ((− 7 ) − 4 ∗ 20 ∗ −6))
(2 ∗ 20)
2
(− − 7 − ((− 7 ) − 4 ∗ 20 ∗ −6))
x=
(2 ∗ 20)
2
Step 3. Use your calculator. Be
sure to put parenthesis around the
entire top of the fraction.
© Copyright 2006 – LeAnn Barr
X= .75 =
2
3
and -.4 = −
4
5
Try this one: Copy the formulas and fill in for a, b and c. Then find the solutions for
this quadratic equation. Be sure to verify by graphing on your calculator.
2. 10 x 2 − 7 x − 12 = 0
a=____, b=____, c=____
(− + (( ) − 4 ∗ ∗ ))
x=
(2 ∗ )
2
x= ________
Algebra I - TEXAS Style
92
x=
(− − ((
x=_______
)2 − 4 ∗
(2 ∗ )
∗ ))
Solving Quadratic Equations
Lesson 8-6
Solve each using the quadratic formula. Make all decimal answers into fractions.
Check your answer by graphing.
3.
4 x 2 + 23 x − 35 = 0
4.
10 x 2 + 41x + 21 = 0
5.
16 x 2 − 66 x − 27 = 0
6.
25 x 2 − 64 = 0
7. Consider the equation
x 2 − 8x + 7 = 0
a. Put in y= and generate a table.
b. Sketch a graph on a grid the size
of the one at the right, identify
specific points on the graph.
c. Where is the line of symmetry,
what does it mean for the graph?
© Copyright 2006 – LeAnn Barr
d. Locate two sets of points affected
by the line of symmetry. (These
are points the same distance
from the line of symmetry, one on
each side of the line of
symmetry.)
e. Where is the equation = to 0 on
the graph how does this relate to
the factors of the polynomial?
Write the factors.
f. What are the dimensions of the rectangle created by this polynomial?
(Remember these are the factors.)
g. Draw and label this rectangle.
h. What is the perimeter of the rectangle created by this polynomial?
Algebra I - TEXAS Style
93
Index
Factoring
Common Term, 82
Difference of Squares, 88
x 2 ± bx + c , 84
x 2 ± bx − c , 86
ax 2 ± bx ± c , 90
Algebra Tiles
Factoring, 82-90
Polynomials, 74-78
Solving Equations, 2
Solving Systems, 60
Constant of variation, 32
Fixed number, 4
Coordinates
Plotting points, 14
Function rules, 26
Functions
Definition of, 13
Function notation, 4, 14
Function rule, 4, 14
Correlation
Definition of, 30
Scatter Plots, 30
Graphing, Inequalities
One variable, 50
Two variable, 52
Application Problems, 54
Dependent variable, 4,14
Direct Variation, 32
Graphing, Linear Equations
Equations of lines, 38
Slope, 34
Slope and a point, 36
Distance between points, 44
© Copyright 2006 – LeAnn Barr
Distributive Property, 78
Domain
Definition of, 3, 12
Mathematical, 12,14
Reasonable, 4
Equations
Solving one step, 2
Solving two step, 8, 14
Variables on both sides, 22
Exponents
Basic rules, 66
Division, 68
Negative, 70
Graphing Quadratics
Parametric changes
(workbook activity 7-3C)
Roots, zeros, solutions,
82-93
Y-intercepts, 84-93
Graphing, Scatter Plots, 30
Graphing Systems of Equations
Functions and Systems, 22
Using your calculator, 62
Independent variable, 4, 14
Algebra I - TEXAS Style
94
Index
Inequalities,
One variable, 50
Two variable, 52
Application problems, 54
Input, 4
Polynomials
Adding and Subtracting, 74
Distributive Property, 76
Multiplying, 78
Pythagorean Theorem
With the distance formula,
44
Line of Best Fit
Scatter Plots, 31
Linear Equations, Graphing
Equations of lines, 38
Parallel, 42
Perpendicular, 42
Slope, 34
Slope and a point, 36
Slope-intercept form, 38
Standard Form, 46
Vertical Shifting, 40
© Copyright 2006 – LeAnn Barr
Patterns, 4-9
Quadrants, 14
Quadratics
Factoring, 82-91
Quadratic Formula, 92
Range
Linear Equations, writing
Given a graph, 39
Given two points, 39
Parallel, 42
Perpendicular, 42
Standard Form, 46
Using slope and y-intercept,
38
Vertical Shifting, 40
Definition of, 3, 12
Mathematical, 12,14
Reasonable, 4
Rate of Change, 4, 32
Relationship Statements, 2
Roots
Definition of, 4
Quadratic 82-93
Negative Exponents, 70
Scatter Plots
Correlation, 30
TI-83+ directions, 31
Output, 4
Scientific Notation, 69
Midpoint Formula, 44
Parent Function
y = x, 34
y = x2,
(workbook activity 7-3C)
Algebra I - TEXAS Style
95
Index
Zeros
Slope
Definition of, 34
Graphing using, 34, 36
Identifying in equations, 38
Parallel and Perpendicular,
42
Slope-Intercept form of Linear
Equations, 38
y = mx + b, 38
Solutions
Definition of, 4
Quadratic 82-93
Standard Form of Linear
Equations, 46
© Copyright 2006 – LeAnn Barr
Systems of equations
Elimination, 60
Functions, 22
Graphing, 62
Perimeter Problems, 58
Substitution, 58
Tables
Using with functions, 14
Using with patterns, 4
x-axis, 4
x-intercept, 4
y-axis, 4
y-intercept
Definition of, 4
Identifying in equations, 38
Algebra I - TEXAS Style
96
Definition of, 4
Quadratic 82-93