Download Practice 03.07: Writing Linear Equations

Algebra 1B Notetaking Guide
SEGMENT 1
This Notetaking Guide contains lesson-by-lesson outline for each module that allows you to take notes on and
review the important concepts. Successful completion of this guide is a required assessment of this course.
Module 1: Algebra Building Blocks
Lesson 1.03 Exponents and Order of Operations

I can simplify and evaluate expressions.




Power
Base
Coefficient
Exponent

Zero exponent

Negative exponent

Scientific Notation

Order of Operations
Rules for Order of Operations
1.
2.
3.
4.
Complete the example:
Simplify: 81 – 3^2 • 2 ÷ (12 - 9)
Lesson 1.04 Variables, Expressions and Equations


I can use variables and constants to write algebraic expressions.
I can translate a verbal expression into an algebra expression or equation and then solve.

Variable

Algebraic expression

Equation

Open sentence

Solution
Complete with which verbal expressions mean the following operations:
Addition
Subtraction
Multiplication
Division
Equals
Write an algebraic expression for each verbal expression:
1.
a number less 5
4.
a number divided by 6
2.
four times a number
5.
twice a number plus 4
3.
the sum of 10 and a number
6.
three times the sum a number and 5
Write a verbal expression for each algebraic expression:
7.
x+4
9.
4(x-3)
8.
2n-6
10. 3-x/4
Evaluate each expression when a = 3, b = 12 and c = 4
11. 3b - ( a + c2 )
12. 2a +3b ÷ c
Define the variable and write an equation:
13. You are 6 years older than your brother. Write an equation expressing your age in terms of your
brother's age. Define your variables.
14. A mechanic charges you $50 to diagnose a problem with your car and $35 per hour to fix it.
Write an equation for the total cost of repair (c) in terms of hours spent (h). What is your total
cost if the mechanic spent 4 hours fixing your car?
Lesson 1.05: Real Number System
& Absolute Value


I can classify, identify and compare real numbers.
I can simplify absolute value.

Real numbers

Irrational numbers

Rational numbers

Integers

Whole numbers

Counting numbers

Absolute value
The symbols used for ordering numbers are as follows:
>
<
<
>
=
greater than
less than
less than or equal to
greater than or equal to
equal to
Name the set(s) to which each number belongs:
1.
-4
2.
3.
0
4.
-6.345
Insert an ordering symbol to make each statement true:
6.
-8____-11
7.
4 ____
8.
___ 0
Find the absolute value:
9.
10. | 0 |
11. | 3.5 |
Lesson1.06:
Properties of Real Numbers

I can apply the associative, commutative and distributive property when solving equations.

Distributive property

Commutative property

Associative property
State the property used.
1.
4(11 + 3) = 4 • 11 + 4 • 3
2.
0 + 7 = 7
3.
4(ab)=4a(b)
4.
6 + b = b+ 6
5.
7 • 1 = 7
Simplify and give a reason for each step.
5x -2(4+x)
Lesson 1.07: Quiz
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
Lesson 1.08:

Solving Linear Equations
I can solve a linear equation.
Solving multi-step equations:
1.
2.
3.
4.
5.
6.
Multiply on both sides to clear the equation of ______________ or decimals. (This is optional but can
make the rest of your procedure easier!)
Use the ________________ law to remove parentheses if they occur.
Collect ___________________ and simplify on each side of equal sign.
Use the addition or subtraction property of equality to get the term(s) with the _________________
alone on one side of the equal sign.
Collect ___________________ and simplify on each side of the equal sign. Use the _________________ or
division property of equality to get the variable alone on one side of the equal sign.
Check your possible solutions in the original equation.
Complete these Practice problems
Solve the following two-step linear equations and check your result.
3.
4. -2x - 1 = -17
Solve the following multi-step linear equations and check your result.
6.
8x - 5x = -15
7.
5(3x + 2) = 40
Solve the following linear equations with variable on both sides and check your result.
8.
8x - 1 = -4x + 23
9. 5(x + 4) = 7(x - 2)
Solve the following special linear equations and explain how many solutions each has.
10. 2(x+3) - 1 = 2x + 5
11. 4x + 2 = 4x - 3
Lesson 1.09:




I
I
I
I
can
can
can
can
define
model
model
model
Writing Linear Equations
a variable in terms of another variable.
perimeter problems.
consecutive integer problems.
distance = rate * time problems.
Step 1: Read the problem carefully and write down what you know and what you are asked to find.
Step 2: Define the variables. You may need to draw a figure, a diagram, or make a chart to do this.
Step 3: Write an equation using your variables.
Step 4: Solve the equation by substituting the values you know.
Step 5: Answer the question posed. Sometimes you have to use the solution of your equation to find the
answer to what is being asked.
Step 6: Check your solution. Be sure your answer makes sense
Complete the following Practice problems:
Practice 01.09 Writing Linear Equations
1.
You received $50.00 from your aunt for your birthday. You want to spend it on some new CDs.
Best Buy has CDs on sale for $12.50. How many CDs can you buy with your birthday money?
Write an equation you can use to find out how many CDs you can buy with $50. Let c equal the
number of CDs you can buy.
2.
At last night's basketball game Sam was the top scorer for his team. The total points scored by
his team was 8 more than twice Sam's personal score. If the final score for his team was 56
points, how many points did Sam score? Let: s = Sam's points and let = team's final points.
3.
Your cell phone plan is $45 per month for 350 minutes and $.12 per each minute used over 350
minutes. If your total bill is $63, how many minutes over 350 did you use? Let t = total bill, p =
monthly plan and m = minutes over 350.
4.
The length of a rectangular garden is 4 yards more than twice its width. The perimeter of the
garden is 44 yards. What are the length and width of the garden?
Let P = 2L+2W
L = length of the garden
W = width of the garden
and L = 2W+4.
5.
The sum of three consecutive integers is - 42. What are the 3 integers? Let n = 1st integer.
6.
Two friends were born in consecutive odd years. If the sum of these two years is 3980, in what
years were they born? Let n = 1st birth year.
7.
Two airplanes depart at the same time from an airport terminal traveling in opposite directions.
The second plane is traveling 150 miles faster than the first plane. After 1 hour they are 850
miles apart. Find the speeds of the two planes.
Lesson 1.10:
Writing and Solving Inequalities


I can solve and graph an inequality.
I can translate and solve real world problems using inequalities.


Inequality
Solution set
Copy an example for each topic below:
Solutions to Inequalities
Solving Inequalities
Graphing Inequalities
Writing Inequalities
Lesson 1.11:
Quiz
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
Lesson 1.12:


Using Formulas and Literal Equations
I can solve a formula for one variable in terms of the other variables.
I can write and use real world formulas.
Copy the Practice problem solutions:
Solve each equation for the given variable.
1.
Solve the perimeter of a rectangle P = 2L + 2 W for L, the length.
2.
Solve ax - y = c for x.
3.
Solve C = 2( )r for r.
4.
Solve 5x + 4y = 4 for y. Find the value of y when x = 4.
5.
Solve the volume of a rectangle V = l w h for w the width.
6.
Solve the Celsius temperature formula
temperature when C = 25o.
Lesson 1.13:
for F, Fahrenheit temperature. Find the
Solving Absolute Value Equations

I can solve equations with absolute values.

Absolute value
Remember that you will be solving absolute value equations for 2 solutions, or no solution. Be sure and check
your solutions in the original absolute value equation.
Complete the Examples below:
Examples
|x - 4| = 12
Explanations
Using the above rule, write two equations and
solve for x.
Add 4 to each side of both equations.
Simplify.
Example 2
Explanation
5 - 4|2x + 1| = -7
First you must ISOLATE the absolute value.
In this problem, we will do that by subtracting 5
from both sides.
Simplify.
Divide by -4.
You have isolated the absolute value.
Now you can write your 2 equations and solve for x.
Lesson 1.14: Practice Test
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
Module 2:
Lesson 2.02:
Functions Review and New
Independent and Dependent Variables


I can determine if a variable is independent or dependent.
I can write and evaluate an equation using independent and dependent variables

Independent variable

Dependent variable
Complete Practice 02.02: Independent and Dependent Variables
Activity 1:
Determine which are independent variables and which are dependent. The variables are underlined.
1.
The price of gas per gallon determines how much money you need to fill your gas tank.
2.
The cost of mailing a package to a certain zip code is determined by the weight of that
package.
Activity 2:
Stephanie must complete a class project for mathematics. The project is worth 100 points if she turns it
in on time. If she turns the paper in late the teacher deducts 5 points for each day the paper is late.
1.
Write an equation for this situation. Let x be the number of days the project is late, and let y be
the maximum number of points that Stephanie can earn for a late project.
2.
Which variable is the independent variable? Which variable is the dependent variable?
3.
Create a table to show the number of possible points for the project if it is up to 7 days late.
Days
Late
Possible
Points
Days
Late
0
6
1
7
2
8
3
9
4
10
Possible
Points
5
4.
How many days late can Stephanie turn in the project and still possibly receive a score of 65?
Use words and mathematics to show your answers.
5.
Stephanie turned in the project 2 days late. Aside from turning in the project late, Stephanie's
the teacher deducted 8 points for not following the directions. What did Stephanie receive on the
project? Use words and mathematics to show your answers.
6.
Using your table from #3 above, graph the teacher's grading scale for turning in the
mathematics project late.
Lesson 2.03:




I
I
I
I
can
can
can
can

Rule

Table

Graph

Input values

Output values

Vertical line test
Definition of a Function
define a function.
determine if a data set or graph is a function.
write a function.
map a function.
Complete Practice 02.03: Definition of a Function
Activity 1:
The First Class Mail rate in the United States for letters, flats, and parcels not over 13 ounces is $0.37
up to the first ounce, and $0.23 for each additional ounce.
1.
Write a function rule for mailing a first class parcel up to 13 ounces. Label the variables.
2.
Create a table to show the cost for mailing an article up to 13 ounces. Write the function rule
above the table. The first column of the table lists the numbers to be substituted for the variable
in the function. The second column lists the values obtained by substituting the numbers in the
first column.
3.
Which variable is independent and which variable is dependent?
4.
Graph your ordered pairs from the table.
Activity 2:
Check your understanding of the vertical line test.
1.
Use the vertical line test to determine which graph is a function. Explain why or why not.
b.
a.
c.
2.
Can the graph of a horizontal line be a function? Explain why or why not.
3.
Can the graph of a vertical line be a function? Explain why or why not.
4.
Is your graph from Activity 1 a function? Explain why or why not.
Lesson 2.04:
Domain and Range

I can determine the domain and range of a relation.
I can map the domain and range of a relation.
I determine if the relation is a function by looking at the domain and range using ordered pairs, a
table or mapping.
I can graph the domain and range and determine if it is continuous, discontinuous, or discrete.

Domain

Range

Function

Continuous function

Discontinuous function

Discrete function



Complete Practice 02.04: Domain and Range
Activity 1:
1.
For each of the following, state the domain and range. Is the set of ordered pairs a function?
Why or why not?
1.
2.
2.
{(2, -3), (4, 6), (3, -1), (5, 5), (2, 3)}
{(4, -2), (-2, 6), (2, -2), (4, 6), (-1, 3)}
Use a mapping diagram to show whether this relation is a function. Label the domain and the
range. Is it a function? Explain in words.
{(0, 2), (3, 1), (-3, -1), (5, 3), (3, -3)}
3.
A taxi driver charges $3.00 for the first mile and $0.75 each mile thereafter.
1. Make a table showing the cost of riding in a taxi per mile up to 5 miles.
2. Use your table to graph the ordered pairs.
c.
What is the domain and range for traveling in a taxi cab for up to 10 miles?
Activity 2:
A moderately active person needs to consume approximately C=15 x p calories per day per pound of
body weight to maintain body weight p in pounds. Moderately active means getting at least 30 minutes
of physical activity a day in some form of exercise such as walking at a brisk pace, climbing stairs, or
active, physical gardening. To lose 1 pound a person must burn 3500 calories. So, to lose one pound in
a week a person needs to lower his/her weight maintenance calories by 500 calories per day.
1.
Using the formula C=15p, make a table of weight maintenance for 100, 110, 125, 130, 150
pounds. Label the table and title the table.
2.
What is the dependent variable and what is the independent variable?
3.
Using your table make a graph for weight maintenance of a moderately active person. Label the
axes. Is this a continuous, non-continuous or discrete function or relation? Explain your answer.
4.
What is the domain and range of your graph? You can express this answer using words.
5.
Write a rule for finding calories per day and trying to lose 1 pound per week. Is this a function?
What is the domain and range of this rule? You may have to graph this to answer the questions
Lesson 2.05: Quiz
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
Lesson 2.06:


Representing Relationships
I can represent relationship numerically, graphically, algebraically and verbally.
I can interpret and analyze relationships in a graph.
Complete Practice 02.06: Representing Relationships
Activity 1:
The table below shows a relationship between x and y.
1.
Which of the graphs - A, B, C, or D represents this relationship?
2.
What is the domain and what is the range of this graph?
3.
Write an equation for the table and graph.
4.
Is this a function? Explain.
5.
Can you describe a situation or relationship that this graph could represent? Be creative!
Activity 2:
Look at the 4 graphs below. Each one represents a different relationship.
1.
A.
C.
B.
D.
Match each graph to one of these situations.
a.
The length of the grass in your yard this summer per week.
b.
Your puppy's weight.
c.
The distance from the ground while riding a roller coaster.
d. The temperature this weekend
4.
Give each of the graphs a title and label the axis.
5.
Sketch a graph to represent your speed as you bicycle on a flat road for 1 mile and then bike
up a hill for 1/2 mile, back down the hill for 1/2 mile and then 1 mile on a flat road again.
Give this a little thought!
Lesson 2.07:
Functional Notation and Rules


I can write a function rule given a table.
I can write a function rule given a real world situation.

Function notation

Function rule
Complete the Examples below:
Look at the 3 tables below. We cannot call each of these functions f(x), because then we could not
distinguish which was for Table 1, Table 2 or Table 3. Let's let Table 1 be defined as _________,
Table 2 as _________
and Table 3 as _________. Now we can write the rule for each and distinguish
one from the other.
Table 1
The function for Table 1 is:
f(x) = _____________
Table 2
The function for Table 2 is:
g(x) = ____________
Table 3
The function for Table 3 is:
h(x) = ____________
We can show each of these functions on the same
graph and label them with their function name.
Notice that for multiple functions the input value or x _______________. The output values are different
because the _____________ are different.
Complete Practice 02.07: Functional Notation and Rule
Activity 1:
The table below shows a relationship between x and y.
1.
What are the next two values for y?
2.
Write a rule for the function in the table using function notation.
Activity 2:
Your kitchen oven is not heating and you call an appliance repair shop. The first repair shop you called,
First-Rate Repairs, charges $35.00 for a home visit and $20.00 an hour for repair. There may be an
additional charge for parts if needed. The second shop you called, In Home Repairs, charges $50.00 for
a home visit and $15.00 per hour plus parts if needed.
1.
Write a function rule, using function notation, to show the cost of appliance repair for home
appliance repair for each shop called. Let f(x) represent the function for First-Rate Repairs and
g(x) represent the function for In-Home Repairs.
2.
What is the independent variable and what is the dependent variable for each function rule?
3.
Make a table for each rule showing the input, function rule, and output values for 1 to 4 hours
repair times.
Lesson 2.08:


Evaluating and Interpreting a Function
I can evaluate a function given a specific input value.
I can solve problems involving functions.
You can think of the function rule as a machine where you put in a value (______________) and the machine
produces a value (________________). Think about when you study for an exam. The amount of effort you put
into studying is the input and your grade is the output!
Using the function rule, you can select a value for the input variable (______________) and substitute that into
the equation or rule to find the output (_____________). These two values give you an ____________________or
the function (x, f(x)). This allows you to graph the function or make a ______________ of function values and
interpret the function by analyzing the graph or table.
Complete Practice 02.08: Evaluating and Interpreting a Function
Activity 1:
New York City taxi cabs are cash only and it's a good idea to have small bills because the cabbies
can't usually break anything higher than $20. While cabs are relatively expensive for a single person,
they can actually be a bargain with 3 or more riders. The rates for taxi cabs are as follows:
Initial fare............$2.00
Each 1/5 mile (4 blocks).$0.30 (Note: 1 mile is $1.50)
Each 1 minute idle......$0.20
Night surcharge.........$0.50 (after 8pm until 6am)
Additional riders........FREE
1.
Write a rule in function notation to find the total cost for a fare in miles during the daytime
(without idling time.)
2.
Use that function to find taxi cost from the airport to your hotel in Manhattan. It is a 15-mile
drive from La Guardia Airport to your hotel. You will have to add $3.00 for tolls and $1.00 for a
tip. Remember you need cash! How much do you have to give the taxi driver? Use mathematics
and words to answer.
3.
What is the input variable (independent variable) and what is the output variable in this function?
4.
Use this function to find the cost from your hotel to Radio City Music Hall which is 2 miles. The
Radio City Music Hall show is over at 10:00pm. How much is your taxi ride back to your hotel?
Use mathematics and words to answer.
Activity 3:
Given f(x) = 3x+5, g(x) = 2x-2 and h(x) = x2+3x-2 , find each of the values.
1.
f(2)
2.
h(-1)
3.
f(2)-h(-1)
Lesson 2.09: Quiz
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
Lesson 2.10:


Formulas
I can solve formulas for a specific variable.
I can write and use formulas to solve a problem.
Complete the Reference sheet with missing formulas.
Mathematics Reference Sheet
Area
Triangle
Rectangle
Trapezoid
Parallelogram
Circle
Key
b = base
h = height
l = length
w = width
= slant height
S.A.=surface area
Use 3.14 or
Circumference
d = diameter
r = radius
A = area
C =
circumference
V = volume
for
.
Volume
Total Surface Area
Right Circular
Cone
Square Pyramid
Sphere
Right Circular
Cylinder
Rectangular Solid
Complete Practice 02.10: Formulas
Activity 1:
Solve each of the following formulas for the indicated variable.
1.
C =
d, for d
3.
V = lwh for l
2.
A = lw for w
4.
SA = 2(lw)+2(hw)+2(lh) for h
Using your formulas as solved above, find the variable for the following values.
5.
Find d when C = 6
6.
Find w when A = 54 and l = 9
7.
Find l when V = 108, w = 3, h =
4
8.
Find h when SA = 158, w = 3 and l = 8
Activity 2:
Garcia's packaging company is designing a new package for oatmeal. The company currently is using a
rectangular box with dimensions 6" x 3" x 8" but wants to repackage this in a right circular cylinder
container that would hold the same quantity of oatmeal as the rectangular box.
1.
Find the volume of the rectangular box.
2.
The volume of a right circular cylinder is V =
r2h. Solve this equation for r.
3.
Using the volume for the rectangular box and h = 8", find the radius for the circular cylinder
container.
Lesson 2.11: Practice Test
Oral Assessment
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Email or call your teacher to set up and
complete your oral assessment PRIOR to taking the Module 2 test.
Module 3: Linear Equation Review and New
Lesson 3.02 Patterns of Change




I
I
I
I
can find the rate of change of table.
can solve a direct variation.
interpret data and write an equation for a direct variation
can solve proportions using direct variation.

Direct variation

Constant of variation
Graphically this means:
As the number of cookies you
eat _____________, so does the
total number of calories. The
constant of variation, k, is
_______________.
As air temperature ____________,
so does the water temperature.
The constant of variation is
The amount that you owe the
bank ____________, while the
number of months in____________.
The constant of variation is
and k is positive.
General Steps for Solving a Direct Variation Word Problem:
1.
2.
3.
4.
Example:
If y varies directly as x, and y = 28 when x = 7, find x when y = 52.
Lesson 3.03:
Slope and Rate of Change


I can find the rate of change using a graph and use it to solve problems.
I can find slope.

Rate of change

Slope of a line

Positive slope
and k is _______________.

Negative slope

Slope formula
Complete Practice 03.03: Slope and Rate of Change
Activity 2:
The graph below shows maximum speeds over approximate quarter-mile distances for 4 different animals.
One exception - which is included to give a wide range of animals - is the elephant, whose speed was
clocked in the act of charging.
1.
Select two points on the line of ground speed for each animal. List those points by animal.
2.
For each animal count the rise and run for your selected points for each animal. List the rise
and run for each by animal.
3.
Determine the slope of the line for each animal using the rise/run. List each slope by animal.
4.
Explain what these slopes mean.
Activity 3:
Find the slope of the line containing the following two points.
1.
(1,3) and ( 2,5)
2.
(-3,1) and (5,4)
3.
(-2, -4) and (-3, -1)
4.
You buy a truck for $20,000. The value depreciates each year you own it. After owning the truck
for 5 years you sell it for $8000. Assume linear depreciation of value and find the rate of
change of the value (this is the slope!). Hint: You are given two points (years owned, $ value of
truck) which are (0, 20000) and (5, 8000). What does this value mean?
Lesson 3.04:
Horizontal and Vertical Lines


I can recognize both horizontal and vertical lines by their graphs and equations.
I can determine the slope and write the equation of a horizontal and vertical line.

H0y =

Vux =
Complete the Examples below:
Horizontal lines have slopes that = zero
Slope
Graph
We know that we can
determine the slope of
a line by using the
formula shown below.
Equation
In order to write an
equation of a
horizontal line, let's
consider the values of
the points on the line.
Here is a t-chart of the
ordered pairs shown on
the RED line to the left.
.
Let's use two of the
points on the BLUE line
to the right. Two of the
points on the line are
(3,6) and (5,6).
M = ___________
Zero divided by any
number is equal to
zero. The slope of a
horizontal line is
______________
Summary facts about horizontal lines
A horizontal line has a slope of _________.
A ____________line is drawn left to right like
the horizon.
The equation of a horizontal line is ________
some number.
All points on a ____________ line will have the
same ______ value.
Notice that the y value
is always -2. The value
of y is not dependent
upon x.
The equation of the
RED horizontal line is y
= -2 because y is
equal to -2 for all
points on the graph.
The equation of the
BLUE horizontal line is
y = 6.
Vertical lines have undefined slope or "no slope"
Slope
Graph
We know that we can
determine the slope
of a line by using
the formula shown
below.
Equation
In order to write an
equation of a horizontal
line, let's consider the
values of the points on the
line.
Here is a t-chart of the
ordered pairs shown on the
RED line to the left.
Let's use two of the
points on the BLUE
line to the right. Two
of the points on the
line are (3,1) and (3,5).
m = ______________
You cannot
__________ by zero
so the answer is
undefined. The slope
of a vertical line is
undefined. A vertical
line has ____________
Summary facts about vertical lines
A vertical line has an ____________________.
A vertical line is drawn up and down.
The equation of a vertical line is _________
some number.
All points on a ______________ line will have
the same _______ value.
Notice that the x value is
always -7. The value of x
is not dependent upon y.
The equation of the RED
vertical line is x = -7
because x is equal to -7
for all points on the graph.
The equation of the BLUE
vertical line is x = 3.
Complete Practice 03.04: Horizontal and Vertical Lines
Activity 1:
Graph the lines x = 2 and y = - 3 on the same coordinate system below.
1.
What is the slope of the x axis?
2.
What is the slope of the y axis?
Activity 2:
Match the graphs below with the following slope descriptions: (a). zero slope, (b). undefined slope, (c).
increasing or positive slope, (d). decreasing or negative slope.
1.
2.
3.
4.
Activity 3:
Find the slope of the line through the points and state if the line is horizontal, vertical, or neither.
1.
(4,5), (4,3)
Activity 4:
Determine the value of r for the following:
1.
A line that passes through the points (r, 5) and (-2, -3) and has an undefined slope.
Lesson 3.05: Quiz
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
Lesson 3.06:





I
I
I
I
I
can
can
can
can
can



Slope
Y-intercept
Slope-intercept form:
Slope-Intercept Form of an Equation
find the x-intercept and y-intercept of an equation.
write an equation given the y-intercept and slope.
write a linear equation given a graph.
graph a line given an equation in slope-intercept form.
write the equation for a horizontal or vertical equation given the x- or y-intercept.
Complete the Examples below:
Example 1
Use the slope and y-intercept to graph the equation y = 3x -2.
The equation is written in slope-intercept form. You can determine by looking at the equation that the
slope is 3 and the y-intercept is -2 or the point (0, -2).
Example 2: Write an equation of a line with a slope of The equation y=mx+b
and y-intercept of (0, 4).
becomes _______________________
Example 3: Write an equation of a line given the following graph of a straight line.
1.
Find the slope or m. Select two points on the line, (-3,2) and (3,0). Use the slope formula,
.
2.
Determine the y-intercept_________________________
The equation y = mx+b becomes __________________________
Example 4:
1.
2.
Find the x- and y-intercepts for the line y = -2x + 10
The y-intercept is easy to find since this equation is written in slope-intercept form or y = mx+b
where b is the y-intercept. In this equation b = _______________ and the y-intercept is the point
________________
The x-intercept is the value of x in the above equation when y = 0.
The x-intercept is ______________________
Equations of Horizontal and Vertical Lines
If you know the x- or y-intercept is a vertical or horizontal line, you can write the equation for that line
because you know the slope.
Since horizontal lines have a slope of 0, the equation y = mx+b becomes __________________________
So, if the y-intercept of a horizontal line is 5, then the equation for that line is
__________________________
Since vertical lines have an _________________ we cannot use the slope-intercept form of an equation to
derive the equation. We do know one point, the x-intercept, (a,0). This gives us the equation for a
vertical line! It is just _______________
So if the x-intercept of a vertical line is 5, then the equation for that line is
___________________________________
Lesson 3.07:




I
I
I
I
can
can
can
can

Slope-intercept form of an equation

Point-slope form of an equation
Writing Linear Equations
write an equation of a line when given the slope and a point on the line.
write an equation of a line when given two points.
graph an equation when given the slope and a point on the line.
write an equation given a table of data.
Complete the Practice problems
Practice 03.07: Writing Linear Equations
Activity 1:
The table below compares three temperature scales: Fahrenheit, Celsius, and Kelvin.
Fahrenheit
Celsius
Kelvin
32
0
273.15
86
30
303.33
122
50
323.33
212
100
373.15
The slope of the lines converting Celsius to Fahrenheit, and Kelvin to Fahrenheit is 9/5.
1.
Using the slope 9/5 and a point for (Celsius, Fahrenheit) from the table above, write an equation
for this conversion. Your equation will be of the form y = 9/5x+b where b is the y-intercept, y
represents F, and x represents C.
Activity 2:
Write an equation of the line that passes through the point and the given slope. Write the equation in
slope-intercept form.
1.
(-3,6), m = 2
2.
(2,5), m = 0
Activity 3:
Write an equation of the line that passes through the two points. Write the equation in slope-intercept
form.
1.
(-1,-1), (2,8)
2.
Write the slope-intercept form of the equation of the line shown below.
Lesson 3.08:




I
I
I
I
can
can
can
can




Standard form of an equation
A
B
C
Standard Form of Linear Equations
identify standard form of an equation and determine the x- and y-intercept.
graph a line in standard form using the x- and y- intercepts.
change standard form into slope-intercept form.
change slope-intercept form into standard form.
Complete the notes below:
Finding the x-intercept
To find the x-intercept:
Finding the y-intercept
To find the y-intercept:
Plot the two intercept points.
Draw a line through them.
Complete Practice 03.08: Standard Form of Linear Equation
Activity 1:
Find the x and y intercepts of the following equation.
1.
3x - y = 6
Activity 2:
1.
Can vertical lines be expressed in Slope-intercept form? Why or why not.
2.
Can horizontal lines be expressed in Slope-intercept from? Why or why not.
Graph each equation:
3.
y = -1
4.
x = 3
Activity 3:
Write each equation in standard form:
5.
y = (-2/3) x + 5
Activity 4:
You work in a pet shop that sells dogs and birds. The shop owner counted exactly 24 legs in the shop.
1.
Let x represent the number of birds in the shop. Write an expression representing the number of
bird legs in the shop (assume each bird has 2 legs!)
2.
Let y represent the number of dogs in the shop. Write an expression representing the number of
dog legs in the shop (assume each dog has 4 legs!)
3.
Write a linear equation in Standard form that models this situation.
4.
What is the x- and y- intercepts of the line of this equation?
5.
There are many solutions to this situation, but what do you think is a solution that makes
sense? How many dogs and birds might the shop owner have?
Lesson 3.09:
Scatter Plots from Data


I can interpret data on a scatter plot graph.
I can write an equation for a line of best fit and use it to make predictions.

Positive correlation

Negative correlation

No correlation
If the scatter plot shows a positive or negative correlation then another use for the scatter plot is to be able
to make predictions about data with a line of best fit.
To fit a line to the data points on a scatter plot:
1.
2
3.
4
Complete Practice 03.09: Scatter Plots From Data
Activity 1:
The table below shows the percentage of students who graduated from Mount Seymour High School over
the past 10 years with a GPA (grade point average) of 2.5 or higher.
Year
% Graduating with
2.5 GPA or higher
2002
40
2001
38
2000
42
1999
45
1998
44
1997
47
1996
50
1995
49
1994
52
1993
55
1.
Construct a scatter plot of this data and draw a trend line.
2.
Is the data correlation positive, negative, or no correlation?
3.
Find the slope of your trend line.
4.
Use your slope and a point on the line to write an equation for your trend line.
5.
If this trend continues, how many students will graduate with a GPA of 2.5 or above in the year
2005?
Lesson 3.10:
Quiz
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
Lesson 3.11:




I
I
I
I
Absolute Value Function
can graph an absolute value function given a table of values.
can recognize the “V” shape of absolute value graph.
can find the vertex of an absolute value graph.
understand how vertical and horizontal shifts change the absolute value function.

Absolute value function

Parent function

Vertex

Reflection

Translation
Describe the vertex and which way the 2nd and 3rd function shifted.
y = |x|
y = |x+2|
y = |x|+2
Complete Practice 3.11
Activity 1:
Consider the graph below:
1.
What is the equation of the line that contains the line segment located in the first quadrant? Use
the Slope-intercept form of an equation.
2.
What is the equation of the line that contains the line segment located in the second quadrant?
Use the Slope-intercept form of an equation.
3.
What is different in these two equations? What is the same?
4.
Looking at the two equations above, what do you think the equation for this absolute value
function is?
5.
Does this graph have a line of symmetry? Where?
6.
What is the vertex of this graph?
7.
What would be the equation of a reflection of the above graph?
Lesson 3.12:
Practice Test
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
MODULE 4: SYSTEMS OF LINEAR EQUATIONS & INEQUALITIES
Lesson 4.02:



Relationships between Lines on a Plane
I can graph two linear equations on the same graph.
I can determine if a systems has one solutions, no solution or infinite solutions.
I can identify consistent/inconsistent and dependent/independent systems of equations.
Complete the notes below:
When you graph two systems or equations with two variables on one graph you will find one of three
possible solutions. Complete the missing notes for each.
Intersecting Lines
Parallel Lines
Coincident Lines
If
If
If
If
a
a
a
a
system
system
system
system
of equations has one or more solutions it is called _________________
has no solution, it is called _____________________________
has exactly one solution it is ____________________________
has more than one solution is _____________________________
Complete the Practice 4.02
Graph the systems below and determine whether the system is consistent or inconsistent by graphing. If
consistent state whether the system is dependent or independent.
1.
x+2y = 3 and 2x-y = 1
2.
y = x-3 and y = 2x-4
Lesson 4.03:



Identifying Special Lines without Graphing
I can identify parallel, perpendicular and coincident lines.
I can recognize the relationship between lines without graphing.
I can write equations of a line given that are parallel or perpendicular to a line.

Parallel lines

Perpendicular lines

Coincident lines
Complete each example:
Example 1
Write an equation of a line in slope-intercept form parallel to y = - 4x+2.
Example 3
Find the slope of a line that is perpendicular to 3x - y = 6.
First rewrite the equation in slope-intercept form.
Example 3:
Find the slope of a line that is perpendicular to 3x - y = 6.
First rewrite the equation in slope-intercept form.
Lesson 4.04:
Solving Systems of Equations using GRAPHING


I can write a system of equations in standard form and change to slope-intercept form.
I can solve a system of equations by GRAPHING and understand what the solution represents.

Systems of linear equations

Point of intersection
Complete the notes below:
Equation 1: 2x + y = 8
Equation 2: 3x + 4y = 17
To graph the first equation, 2x + y = 8, we need to identify the slope and the y-intercept. Let's begin by
putting the equation in Slope-intercept form. Remember that the Slope-intercept form of a linear equation
is _______________ where m is the slope and b is the y-intercept.
2x + y = 8
2x - 2x + y = -2x + 8
y = ____________________
The slope is_______ and the y-intercept is ----------------------
To graph the line:
To graph the second equation, we need to identify the slope and yintercept of Equation 2. Let's begin by putting the equation in
____________________
3x + 4y = 17
4y = -3x + 17
Y = ___________________
So the slope is ___________and the y-intercept is ________________or
(0,4.25). To graph the line, you need to go to the y-intercept and
then count down 3 and to the right 4 to find another point on the
line.
Here are the two graphs
on the same coordinate
plane.
The point _____________________ is the point of intersection and therefore the ________________ to the
system of equations. This means that the ordered pair (3,2) will work in __________________.
Answer these questions:
1.
2.
3.
Is the graphing method always reliable? Why or why not?
What if the 2 lines are parallel? What is the solution?
What if the 2 lines are actually the same line (coincidental)? What is the solution?
Complete Practice 04.04: Solving Systems of Equations Using Graphing
Activity 1:
Solve the following systems of equations graphically. Determine whether the system has one solution, no
solution, or infinite solutions. Write your answer as an ordered pair (x,y). Check your solution in each
equation.
1.
2x + y = 9
-x - y = -8
2.
y = -2x + 1
y = -2x - 3
Activity 3:
Lauren makes money by babysitting. She offers parents one of two plans.
Plan 1: A flat rate of $6.00 per hour
Plan 2: An initial fee of $10.00, plus $4.00 per hour.
Graph these two plans and answer the following.
a.
For what number of hours are these two plans equal?
2.
Which plan to you think is most beneficial to Lauren and why?
Lesson 4.05:
Quiz
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
Lesson 4.06:


Solving Systems of Equations using SUBSTITUTION
I can solve an equation for another variable.
I can use the substitution method to solve a system of equations.
Complete the notes below:
Substitution
The objective of the substitution method is
____________________________________________________________________________
Example 1: In a restaurant, two groups placed the orders shown in the table below.
Number of Small Lunch
Plates
Number of
Large
Lunch Plates
Total
Price
Group A
Order
2
1
$15
Group B
Order
3
2
$26
Based on this information, what is the price, in dollars, of a small lunch plate and a large lunch plate?
Let "x" be the number of small lunch plates sold and "y" be the number of large lunch plates sold.
Write an equation for the Group A order and an equation for the Group B order from the given
information.
Group A Order : _________________
Group B Order : _________________
Solve these two equations using substitution to find the number of small lunch plates sold (x) and the
number of large lunch plates sold (y).
1. Solve for one of the variables in terms of the other in one of the equations.
This means that the value of y is ________________________
2.Substitute the value of y into the second equation of the system.
Notice that the ____________ is put in the place of ______. This is called ________________.
Now you can solve the equation to find the value of x.
Now you can solve the equation to find the value of x.
3x + 2 (-2x + 15) = 26
3x - 4x + 30 = 26
- x + 30 = 26
-x = 26 -30
-x = - 4
(-1) (-x) = (-1)(- 4 )
x =______________
3.Substitute the result into one of the equations to find the other variable
To find the value of y, _____________________________________________________
2x + y = 15
2(4) + y = 15
8 + y = 15
y = 15 - 8
y = ________
The solution to this system of equations is the point (_____). This means that the small plate lunch is
$_____ and the large plate lunch is $______.
Complete the Practice 4.06 Problems
Activity 2:
Juan and Alex are both saving money for college. Juan saves $150 per month and currently has $2000
in the bank. Alex saves $250 per month and has only $1000 in the bank. Not counting the interest they
will earn, how many months will it take until both boys have saved the same amount of money?
Let m represent the # of months it takes Juan and Alex to save the same amount of money. Let A
represent the amount of money they will have after m months.
a. Write an equation for the amount of money Juan will have after m months.
b. Write an equation for the amount of money Alex will have after m months.
c. Solve the two equations to determine the number on months it will take them to save the same
amount of money. Your solution will be (m,A).
Lesson 4.07:

Solving Systems of Equations using ELIMINATION
I can solve a system of equations by using the ELIMINATION METHOD (ADDITION)
Complete the notes below:
The objective of the elimination method is to create
________________________________________________________________________. Here is what that means:
Let's solve the following system of equations:
2x - y = 16
x + y = 5
Notice that the y terms have ____________________. One of the y's is positive and the other y is
negative. So when you add the y terms together, you will get zero. They will add out. This is GREAT
news because it allows us to solve for the x variable.
Notice that the y term is gone (eliminated) and we can solve for x by dividing both sides of the
equation by 3. __________________________________________________
A system is not completely solved until values for both _____and ______are found. Do not
stop after finding the value for one variable.
To find the value of y, substitute your ______value into one of the original equations. Let's use 2x - y =
16. We know that x = 7 so here is the work for finding the value of y.
Y = _______
The solution to this system of equations is the point (_______)
Complete the Practice 4.07 Problem (Note the subtraction used)
Activity 3:
Amanda volunteered to sell tickets for the school band concert. The cost of an adult ticket was $2.50
and the cost of a student ticket was $1.00. She sold a total of 475 tickets and collected a total of
$826.
Band Concert Admission Prices
ADULT
$2.50
STUDENT
$1.00
a. Write a system of two equations that could be used to find out how many of each type of ticket was
sold.
Let:
a = the number of adult tickets sold and
s = the number of student tickets sold
b. Solve the system of equations for a and s to determine how many of each type of ticket were sold.
Lesson 4.08:
Solving Systems of Equations using MULTIPLICATION/ELIMINATION

I can multiply one or both equations by the appropriate numbers so that the coefficients of one of the
variables are opposites of each other and will cancel out.

I can solve a linear system by eliminating one of the variables.
Complete the notes below:
Multiplying Both Equations When Using the Elimination Method
Sometimes you have to multiply ______________ by a number in order to get opposite coefficients for
one of the variables. Look at the two equations below.
5x - 2y = 0
2x - 3y = -11
There are no ________________. You will have to make the coefficients of one of the variables opposite
to eliminate a variable.
For the x variable you have ____ and ____. Think about the least common multiple of ____ and ___ ...it
is _____. We could make one of them ___ and the other a ____.
Multiply the first equation by _____and the second equation by -_____ in order to produce opposite
coefficients for x.
2(5x-2y = 0)
-5(2x-3y = -11)
__________________
___________________
Now we can add them.
Remember we are looking for an ordered pair solution, so now you must find _______________
To find the value of x, _________________________________________________.
The solution to this system of equations is the point _______________.
Lesson 4.09:
Quiz
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
Lesson 4.10:




Applications of Linear Systems
I can model rate-time-distance problems using two variables.
I can model age, coin and number problems using two variables.
I module percentage solution problems using two variables.
I can decide which method for solving a system (graphing, substitution or elimination) will work best for
a given problem.
Methods for Solving Systems of Linear Equations
Graphing
Use this method if the equations graph easily and the
_________________________ has integers for
coordinates. You might also use this if you only want
to estimate the solution.
Substitution
Use this method if one of the variables has
_______________ for the coefficient.
Elimination
This system works for any system of linear equations.
Use addition if one of the variables has
______________, otherwise _____________ equation(s) by
a value that produces opposite coefficients and
then_____ the two equations.
Matrices
To be covered later!
Complete Practice 04.10: Applications of Linear Systems
Activity 1:
1. Sunset rents an SUV at $21.95 plus $0.23 per mile. Sunrise rents the same vehicle for $24.95 plus
$0.19 per mile.
Let m = miles
Let c = cost
a.
Write a system of equations that can be used to determine the number of miles for which the
cost at both rental companies is the same.
b.
Determine the number of miles for which the cost is the same.
Activity 2:
2. At the grocery store, oranges are one price and candy bars are another. Susan buys three oranges
and one candy bar for $2.70. Fred buys one orange and 4 candy bars for $6.40.
Let o = the price of one orange
Let c = the price of one candy bar
a.
b.
Write a system of equations that can be used to determine the cost of oranges and candy bars.
Determine the cost for an orange and a candy bar at the grocery store.
Activity 3:
3. A kayaker paddled 2 hours with a 6 mph current in a river. The return trip against the same current
took 3 hours. Find the speed the kayaker would make in still water.
How do I solve this problem?
The distance is the same both upstream and downstream and can be represented by "d." The rate is
tricky. The trip downstream (with the current) is faster than the upstream trip (against the current). We
don't know the speed the kayaker makes in still water, so we can represent that with "r."
We do know that going with the current will
current will subtract 6mph from his rate (r unknowns). Using d = rt again, the equation
equation downstream, and solve the system.
add 6mph to this rate (r + 6), and going against the
6). Now we can write two equations (we have two
upstream is: d = (r - 6)3. You write the equation for the
Don't forget the units in your answer.
Activity 4:
4. Your piggy bank has 25 coins in it; some are quarters and some are nickels. You have $3.45. How
many nickels and quarters do you have?
Hint: Write one equation for the number of coins that you have and a second equation for the value of
the coins. (An unknown number of nickels would have the value of 5n.)
Let q = the number of quarters you have.
Let n = the number of nickels you have.
a.
Write a system of equations to determine the number of quarters and nickels you have.
b.
Solve for the number of quarters and nickels by graphing the system of equations. How many
quarters do you have? How many nickels do you have?
Lesson 4.11:
Graphing Linear Inequalities



I can describe an inequality algebraically and graphically.
I can find the solution to an inequality by graphing.
I can write and solve inequalities for real world problems.

Boundary line
Complete the notes below:
The equation of a line in two variables can be transformed into an inequality. Earlier you learned how to
graph equations of a line using the _______________________. The graphing of inequalities is very similar.
Example 1
Graph -2x + y < 1.
The graphing process is shown in detail below.
Step 1
Graph the y-intercept.
Step 2
Count the slope to find
another point on the
line.
The y-intercept is (0,1).
The slope is 2 so you
count up 2 and to the
right 1.
Step 3
Since the inequality has
a less than or equal to
sign, connect the dots
with a solid line.
Step 4
Since the values of y
are less than the line,
the shading is below the
line. All the solutions to
the linear inequality are
located in the shaded
area. There are an
infinite number of
solutions.
Example 2
Graph x + 2y > 4.
The graphing process is shown in detail below.
Step 1
Step 2
Graph the y-intercept.
Count the slope to find
another point on the line.
Step 3
Step 4
Since the inequality has
a greater than sign
(not equal to), connect
the dots with a dashed
line.
Since the values
of y are greater
than the line, the
shading is above
the line. All the
solutions to the
linear inequality
are located in
the shaded area.
There are an
infinite number of
solutions.
The y-intercept is (0,2).
The slope is
so you
count down 1 and to the
right 2.
Note:
What is the difference between the inequalities and the line of the graph of each inequality?
Complete Practice 04.11: Graphing Linear Inequalities
Activity 1:
Graph each linear inequality.
1.
y < 4x
a.
b.
c.
- 1
What is the boundary line?
Is the boundary line dotted or solid?
Give a possible solution.
2.
-2x + 3y < -6
a. What is the boundary line?
b. Is the boundary line dotted or solid?
c.
Give a possible solution.
Activity 2:
1.
Which linear inequality below describes the above graph?
a.
y <
x -2
b.
y >
x -2
c.
y >
x - 2
d. y <
x - 2
Lesson 4.12:
Graphing a System of Linear Inequalities


I can solve a system of inequalities by graphing.
I can write and solve a system of inequalities representing a real world situation.

System of Inequalities
Solving a System of Inequalities (Copy the 2 steps from Example 1)
1.
2.
Complete Practice 04.12: Graphing a System of Linear Inequalities
Activity 1:
Solve and graph the systems of inequalities below.
1.
5x + 3y > -6 and 2x + y < 6
a. Name three solutions.
b. Name three ordered pairs that are not solutions.
c. Is (2, 1) a solution? Why?
Activity 3:
You received a couple of gift certificates to Edges Bookstore for your birthday in the amounts of $25
and $50. All CDs at the store cost $15 and all paperback books cost $9. You want to buy some books
and at least 2 new CDs for summer reading and listening.
1.
2.
3.
4.
5.
6.
Write a system of inequalities for the number of books and CDs that describe this situation.
Label the variables.
Graph this system to show all possible solutions.
Give three solutions to this system.
Give an ordered pair that is not a solution.
Is (4, 3) a solution? Explain with mathematics.
Find a solution in which you spend almost all of the gift certificate.
Lesson 4.13:
Quiz
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
Lesson 4.14: Practice Test
Oral Assessment
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Email or call your teacher to set up and
complete your oral assessment PRIOR to taking your Module 4 test.
Module 5: Polynomials
Lesson 5.02:
Polynomial Expressions



I can identify a polynomial
I can name the parts or terms of a polynomial.
I can name the degree of a polynomial.

Monomial

Binomial

Trinomial

Degree of a term

Degree of a polynomial

Standard form of a polynomial
Polynomial
6x-3
3x3+4x2-2x+6
7
x2-5x+4
5x3y2
Degree
Number of Terms
Name Using Number of Terms
Lesson 5.03:



Adding and Subtracting Polynomials
I can add polynomials.
I can subtract polynomials.
I can use polynomials to model perimeter and area.
*SAS Curriculum Pathways Username
Complete the notes below:
Collecting Like Terms
You can simplify a polynomial by collecting and combining like terms.
Complete the right side of each step
Addition of Polynomials
Rule: Add the coefficients of the terms that have the same variable.
Example 1
(x2 - 1) + (2x2 - 5x + 4)
3x2 - 5x + 3
Example 2
(x3y - xy + 8) + (3xy - 9)
x3y + 2xy - 1
Subtraction of Polynomials
Rule: Distribute the subtraction sign through the second parenthesis and then add the coefficients of the
like terms.
Example 3
(x + 7) - (4x2 - 3x + 12)
-4x2 + 4x - 5
Example 4
(2ab - 7b2) - (-3ab + 6a2 - b2)
-6a2 - 6b2 + 5ab
Complete Practice 05.03: Adding and Subtracting Polynomials
Activity 2:
1.
The above is a design for a walkway with a statue and fountain surrounded by grass in a local
park.
a. Write a polynomial term to represent the walkway area.
b. Write a constant term to represent the statue and fountain area.
c. Write a polynomial to represent the green grass area. Hint: You will have to find the area
of the green section and subtract the walkway area and statue and fountain area.
d. Can you simplify this polynomial using addition or subtraction? Why or why not?
Lesson 5.04:
Multiplying and Dividing Monomials



I can use the product rule to multiply exponential expressions with like bases.
I can use the quotient rule to divide exponential expressions with like bases.
I can multiply and divide monomials.

Power

Base

Exponent
Rule 1 for Exponents: Product of Powers
am · an=am+n
______________________________________________________
Example 3:
-7xy(8x2y3)
= _________________
Rule 2 for Exponents: Quotient of Powers
_____________________________________________________
Example 3:
Example 4:
Example 5:
Complete Practice 05.04: Multiplying and Dividing with Monomials
Activity 1:
1.
x3(x4)
2.
-12x(4x5)
3.
-5x3y(-4x5y2)
4.
5.
6.
Activity 2:
Apply your understanding of multiplying and dividing monomials to geometric figures.
1.
Find the measure of the areas of the rectangles below. Remember A = lw.
a.
2.
b.
Write the ratio of the area of the circle to the area of the square in the simplest form in the
figure below. If the radius of the circle is 2x what is the length of the side of the square? Your
ratio should be
.
Lesson 5.05:


Laws of Exponents
I can use the power rule to raise powers to powers.
I can raise a product to a power and a quotient to a power.
Rule 1: Product of Powers When
multiplying powers with the same
base, add the exponents.
Example:
2x2y3(3x2) = (2)(3)(x2+2)(y3) = 6x4y3
a
m
· a = a
n
m+n
Rule 2: Quotient of Powers When
dividing powers with the same base,
subtract the exponents.
Example:
Do not confuse addition or subtraction of terms with multiplication and division! Notice there is no law
of exponents for adding or subtracting monomial terms. This is because when we add or subtract
monomials the variables and exponents _________________________!
For example: 7x5+2x5 = 9x5,
but (7x5)(2x5 ) = 7 · 2 x5+5 = 14 x10 .
Rule 3: Power to a Power
When raising a power to another power, multiply
the exponents. (The coefficients are raised to the
power like normal.)
(am)n = am
· n
Rule 4: Product to a Power
When raising a monomial to a power, each factor
of the monomial is raised to the power.
(ab)m = ambm
Rule 5: Quotient to a Power
When raising a division of monomials to a power,
raise both the numerator and the denominator to
the power.
Putting it all together: More complex problems may require that you use both the power to a power rule
and the product rule. Here is an example.
( 2x5)2 ( -3x3)4
Step 1:
Step 2:
( 2x5)2 ( -3x3)4 = (4x10)(81x12 )
(4)(81)(x10+12) = 324x22
Step 3:
Complete Practice 05.05: Laws of Exponents
Activity 1:
Simplify each expression.
1.
(-2x5y)2
2. (-m2n3)4
3.
(5x3y5)2
4. (-2xy)2(3x2y5)3
5.
Lesson 5.06: Quiz
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
Lesson 5.07:

Multiplying a Polynomial by a Monomial
I can multiply a polynomial by a monomial.
A few things to remember when multiplying terms in polynomials.
Coefficients are _____________ together
Exponents are_____________
Example of monomial x monomial:
2x3(-4x6) = -8x9
Complete the Examples from the lesson:
Example 1
Example 2
Some expressions may contain like terms. If so, you will want to simplify by combining like terms.
Example 3
Find x(3x-5) + 3x
Remember you are using the distributive property and multiplication of exponents with like bases property
to multiply a monomial and a polynomial.
Lesson 5.08:
Multiplying Polynomials


I can use FOIL to multiply polynomials that are binomials.
I can multiply any two polynomials by using the distributive property.




F
O
I
L
Complete Example Multiplying Two Binomials
(2x - 1)(x + 5)
Multiply the 2x from the first binomial with both terms from the second binomial. You will get 2x 2+10x.
.
Now, multiply the -1 from the first binomial with both terms from the second binomial. You will get -x-5.
(Pay attention to the minus in front of the 1 when multiplying here. It multiplies with both the x and the
+5 from the second binomial.)
Final answer ____________________________
Polynomials (many terms) times Polynomials (many terms) Reminders:



x(x) is ________. Since you are multiplying, you must ADD the exponents. x(x 6) is x7. Since you are
multiplying, you must _______ the exponents.
2x plus 3x is _______. Since you are adding like terms, only the ________ will change, NOT the
exponents.
Terms must be _______ to add them. 4x2 and 3x5 would ______ be like terms because the powers
are not the same.
Complete Practice 05.08: Multiplying Polynomials
Activity 1:
Multiply each of the following polynomials.
1.
(x + 5)(x +6)
2.
(4x - 5y)(3x + 2y)
3.
(2x + 3)(2x2 + 3x - 1)
Lesson 5.09:


Special Products
I can use patterns to solve the square of a binomial.
I can use patterns to solve the sum and difference of a binomial.
(binomial)2
F- first terms
O-outer terms
I-inner terms
(x+5)2 = (x+5)(x+5)
(x+2)2 = (x+2)(x+2)
(y+7)2 = (y+7)(y+7)
(x-5)2 = (x-5)(x-5)
(x-2)2 = (x-2)(x-2)
(y-7)2 = (y-7)(y-7)
(a+b) = (a+b)(a+b)
Squaring a Binomial
(a + b)2 = a2 + 2ab + b2
Multiplying the Sum and Difference of the Same Terms
(a + b)(a - b) = a2 - b2
(a - b)2 = a2 - 2ab + b2
L-last term
F+O+I+L
Complete Practice 05.09: Special Products
Activity 1:
Find each product by applying the rule for squaring a binomial.
1.
2.
(x - 4)2 =
(4a - 5b)2 =
Activity 3:
Find the product by applying the rule for multiplying the sum and difference of two terms.
1.
2.
(x-3)(x+3) =
(4a + 5b)(4a - 5b) =
Activity 4:
1.
Find the area of a square with side, s = x+3.
Lesson 5.10:
Quiz
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
Lesson 5.11:
Dividing Polynomials

I divide a polynomial by a monomial

Review all the words in the lesson on polynomials
Division of a polynomial (many terms) by a monomial (one term only)
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and write
each quotient in lowest terms.
Example 1
Complete Practice 05.11: Dividing Polynomials
Activity 1:
1.
=
2.
Activity 2:
The expression 16x2+28x-36 represents the perimeter of a square. Which of these expressions represents
the length of one side?
a. 8x2+14x-18
b. 2(16x2+28x-36)
c.
d. 4x2+7x-9
Lesson 5.12:
Scientific Notation

I can write a number in standard form or scientific notation form.


Scientific notation
Standard form
Rule 6: Zero as a Power
When raising a base to the zero power, your
answer will always be 1. (The base, a, cannot
also be zero.)
a0 = 1
Rule 7: Negative Exponents
When raising a base to a negative power, the
expression can be written as a fraction. The base
will be written on the reciprocal side of the
fraction with a positive exponent.
Writing numbers in scientific notation
Step 1: _______________________________________________________________
Step 2: _______________________________________________________________.
The power of 10 corresponds to the number of places that the decimal has to move to get back to
where it was in the original number.
If the decimal needs to be
powers of 10 indicate that
If the decimal needs to be
powers of 10 indicate that
moved to the right, you will use a positive exponent. Positive
you are multiplying to make the number larger and larger.
moved to the left, you will use a negative exponent. Negative
you are dividing to make the number smaller and smaller.
Writing numbers in standard notation
Step 1: __________________________________________________________________
The power of 10 tells you how many spaces to move the decimal. The sign of the power tells you which
direction to move the decimal.
Positive exponents mean we must move the decimal to the right. This indicates the
number is larger than it looks in scientific notation.
Negative exponents mean the decimal must move to the left. This indicates the number is
smaller than it looks in scientific notation.
Step 2: _____________________________________________________________________
Complete Practice 05.12: Scientific Notation
Activity 1:
Write each number below in scientific notation.
1. 3,000,000
3. 0.0000652
Write each number in standard notation.
5. 2.5 x 106
8. 3.2 x 10-2
Activity 2:
Solve each problem.
1.
2.
3.
A virus is 1.3 x 10-6 meters long. Write the length of the virus as a decimal.
The surface area of earth is approximately 510,000,000 km2. Write this value in scientific notation.
A scientist is comparing the weights of the four molecules listed in the table below.
WEIGHTS OF MOLECULES
Molecule
Weight (in kilograms)
Salt
9.3550 x 10
Pure Water
2.879 x 10
-26
Hydrochloric Acid
6.832 x 10
-22
Potassium hydroxide
8.976 x 10
-24
-28
Which of these molecules is the heaviest?
Lesson 5.13:
Practice Test
Math topic(s) ______________________________________
Score % _____________
Copy the problems you got wrong with the correct solutions. Contact your teacher to review any of the
problems especially if your score is below a 70%.
ALGEBRA 1BSEGMENT 1 FINAL EXAM