ALGEBRA I
[pic]Section 1.1 Variables in Algebra
Warm-up/Bonus Problems
A. Write 58% as a decimal and as a reduced fraction.
B. Evaluate each expression when x = 5 and state the basic operation being performed.
i) x + 12 ii) 30/x iii) 43 – x iv) 3x
C. Simplify: 5 + 16(2) – 11
D. Find the perimeter and area of the rectangle.
E. Complete each statement using < or >.
i) 399_____309 ii) 29.03_____29.3 iii) 5010_____5001
I. Strategies for Reading
Every chapter begins with a Study Guide (see p. 2).
What four features are included in this Study Guide?
1_______________________________________
2_______________________________________
3_______________________________________
4_______________________________________
*Note: Doing the Skill Reviews for each chapter may help you with the Warm-up/Bonus Problems.
Other features of this book include:
(As you look at each of the following, reflect on their usefulness.)
STUDY TIP p.17
SKILLS REVIEW p. 17&21
LOOK BACK p. 50
EXTRA PRACTICE p. 12
HOMEWORK HELP Examples in book (p. 12) as well as the Internet (p.13)
KEYSTROKE HELP p. 15
II. Vocabulary
A _______________ is a letter that is used to represent one or more numbers
The numbers are the _______________ of the variable.
A ___________________________ is a collection of numbers, variables and operations.
Example 1: Write a variable expression using the numbers 2 and 3 and the
variables x and y.
Example 2: Evaluate the following if x = 3 and y = 5.
a) [pic] b) 2.8x – 1.5y c) 10xy
d) [pic] e) 3(2x + y)
Applications
A. Average Speed = Distance/Time = d/t
Find the average speed of a car that traveled 102 miles in 3 hours.
B. Simple Interest = Principal(Rate as a decimal)(Time in years)
If you deposit $1000 into an account earning 7% per year,
how much simple interest will you earn after 6 months?
C. Perimeter & Pythagoreaous
Find the perimeter of the following triangle.
Hint: Use Glossary p. 847 or Math Reference Sheet (back cover).
Section 1.2 Exponents and Powers – Day 1
I. Vocabulary
An expression like [pic] is called a ________________. The ____________ 4 represents the number of times the _______________ 5 is used as a factor.
What is a factor?______________________________________
Example 1: Evaluate each expression for t = 9.
[pic]
What pattern do you see for the last digit of each answer?
Example 2: Evaluate each expression when x = 5 and compare your results.
[pic]
Example 3: Given a = 3 and b = 4
Is [pic][pic]
______________________ indicate which operation should be done 1st.
II. Application
Example 4:
Find the surface area and the volume of a cube with edge length = 3 cm.
What are the units of measure for each?
Example 5 Using a Table
Show the relationship between the side length of a square, its perimeter and its area by completing the table below.
|side |1 cm |2 cm |3 cm |4 cm |5 cm |
|perimeter | | | | | |
|area | | | | | |
Area is a _______________ unit. Side length and perimeter are _______________ units or units of _______________.
Example 6: Volume
A fish tank has the shape of a cube. Each edge is 4.5 feet long.
a) Find the volume in cubic feet.
b) How many gallons of water will the cubic tank hold?
(1 [pic]= 748 gallons)
Section 1.3 Order of Operations
I. Warm-up/Bonus Problems
Simplify each expression.
[pic]
II. GEMDAS
Order of Operations
G ___________________________________________________
E ___________________________________________________
MD _________________________________________________
AS __________________________________________________
Example 1: Evaluate each expression when x = 6.
[pic]
Example 2: Identify grouping symbols and simplify the following.
[pic]
Example 3: [pic] _______________
[pic]
[pic]
Insert grouping symbols into the problem above to produce the indicated values.
i) 146 = __________________________
ii) 8 = ______________________________
iii) 14 = __________________________[pic]
III. Applications
Example 4 Calculating Sales Tax
While spending the day at the mall, you bought two shirts that cost $35 each and a computer game for $45. If the tax on clothing is 3% and the tax on the computer game is 8%, what keystrokes would you enter into your calculator to show the correct amount you owe for the three items?
Section 1.4 Equations and Inequalities
Vocabulary
An _______________ is formed when an equal sign (=) is place between two expressions. In the equation 5x – 9 = 21 _________ is the right side and _________ is the left side.
An equation that contains one or more variable is an __________________.
Example 1: Finding Solutions to Open Sentences
A. Check whether the numbers 1, 2 and 3 are solutions of the
equation 2x + 3 = 5.
B. Check whether the numbers 3 and 4 are solutions of the
equation 5x – 7 = 8.
Example 2: Using Mental Math to Solve Equations
1. Match the equation with the question that can be used to find a solution of the equation.
2. Then use mental math to solve the equation.
Equation Mental Math Question
1.) x + 2 = 6 A.) 2 times what number gives 10?
2.) x – 3 = 4 B.) What number divided by 3 gives 1?
3.) 2x = 10 C.) What number minus 3 gives 4?
4.) [pic] = 1 D.) What number cubed gives 8?
5.) x3 = 8 E.) What number plus 2 gives 6?
II. Inequalities are another type of Open Sentence.
For each inequality symbol, state its meaning
Symbols < > [pic]
Meaning ____________ ___________ ____________ ___________
Example 3: Decide whether 3 is a solution of the inequality.
a.) [pic] b.) [pic] c.) [pic][pic][pic]
Example 4: Tell whether the given number is a solution of the inequality.
a.) [pic]; 6 b.) [pic]; 7
Example 5: Special Angles
A. An obtuse angle is an angle whose measure is between 90 and 180 degrees. If the measure of an obtuse angle is 2y + 3, what are the possible values of y?
B. An acute angle is an angle whose measure is between 0 and 90 degrees. If the measure of an acute angle is 3x, what are the possible values of x?
Example 6: Multi-Step Problem
You are saving money to buy a stereo. You need at least $300 to pay for the one you want to buy, and you have already saved $125. You can save an additional $10 every week.
a. Write an inequality to model the situation using w for weeks.
b. What do the 125 and 10 represent?
c. How long will it take you to save enough money?
d. If you only save $5 a week, how long will it take to save enough money?
Section 1.5 A Problem Solving Plan Using Models
I. Translating Verbal Phrases
Operation Verbal Phrase Expression
Addition
Subtraction
Multiplication
Division
Example 1: Translate the phrase into an algebraic expression.
a. Six less than 4 times a number b.) Three more than the difference
of five and a number n
c.) A number y decreased by the sum of 8 and the square of another number x
II. Writing an Algebraic Model
Example 2:
You and your friends go to a video store to buy DVDs on sale for $12.50 each. Together, you buy 6 DVDs and you spend $79.50, which includes tax. Use mental math to solve the equation for how much tax you paid.
Verbal Model:
Labels:
Algebraic Model:
A Problem Solving Plan Using Models
VERBAL MODEL: Ask yourself what you need to know to solve the problem. Then write a verbal model that will give you what you need to know.
LABELS: Assign labels to each part of your verbal model.
ALGEBRAIC MODEL: Use the labels to write an algebraic model based on your verbal model.
SOLVE: Solve the algebraic model and answer the original question.
CHECK: Check that your answer is reasonable.
Example 3: Using a Verbal Model
You are running in a marathon at a speed of 5 miles per hour. After 3 hours, you have run 15 miles. If you maintain your current speed, how long will it take you to run the last 11.2 miles of the marathon?
Example 4: A salesperson drives at a speed of 50 miles per hour. When he is 175 miles f rom his destination, he remembers he has a meeting in 3 hours.
a. At his current speed, will he be on time for his meeting?
b. At what minimum speed should he travel from this point on if he wants to be on time for the meeting?
Example 5: Write an algebraic model and solve each of the following problems.
A. Six friends went to a restaurant for dinner. The waiter gave them a bill for $98. At the register a tax and tip of $22 was added to the bill. How much did each person contribute to pay an equal share of the bill?
B. How fast must one travel to go 100 miles in 2.5 hours?
C. The perimeter of a square is equal to four times the difference of the length of the side and two. If the perimeter is 20 meters, what is the length of the side of the square?
Section 1.6 Tables and Graphs
I. Vocabulary
____________ is plural and refers to information, facts or numbers that describe something.
Where in everyday life do we interpret data?
Oftentimes we use a ____________ or ___________ to organize data.
For what purpose do we organize data?
II. Example 1: Tables
The data in the table shows the number of endangered species of animals in the United States as of August 31, 1998.
A. Which group has the least number of
endangered species? Explain how you
Know.
B. Which group has the greatest number of
endangered species? Explain how you
know.
C. Suppose the names of the groups are
Listed in alphabetical order. Would
questions A and B be easier or harder
to answer? Explain.
Example 2: Bar Graph
The following bar graph represents the number of worldwide shipments of personal computers, in millions.
[pic]
a. During which 2-year period did the number of shipments increase the most?
b. Is the above graph misleading? Explain.
c. How would beginning the number of PCs at 20, in other words, chopping off the bottom of the bar graph make the graph misleading?
Example 3: Line Graph
The line graph below has been created using the same data from example 2.
[pic]
a. How can the line graph be used to answer (a) from above?
b. Is there ever a 2-year period where the number of worldwide shipments of personal computers decreased?
Section 1.7 Functions
I. Vocabulary
A ____________ is a rule that establishes a relationship between two quantities called the ____________ and the ____________.
**For each input, there is __________ ______ output.**
**More than one input can have the same output.**
Input values are also called the ____________ of the function and output values are called the ____________ of the function.
Example 1: Given y = 2x, find the values of y for values of x = 1, 2 and 3 using a input-output table. Is it a function?
Example 2:
a. Make an input-output table for y = x2 using x = 0, 1, 2, and 3.
b. Does the table represent a function? Justify your answer.
c. Describe the domain and the range.
Example 3: You bicycle 4 miles and decide to ride for another 2.5 hours at 6 mi/hr. The distance you have traveled d after t hours is given by
d = 4 + 6t, where [pic]
a. For several inputs t, calculate an output h and make an input-output table.
b. Make a line graph of the data.
Example 4: Writing an Equation
To fix a car, a mechanic estimates that it will take 2 to 5 hours of work and $270 in parts. The mechanic charges $35 per hour. Represent the total cost of the repair C as a function of the hours h that it takes the mechanic to fix the car for every hour starting with 2 hours and ending at 5 hours. Write an equation for the function.
Use the equation to make an input-output table for the function.
Example 5: Trip to the Fabric Store
a. You are buying fabric that costs $6.40 per yard. Write an equation for the total cost of the fabric C as a function of the yards of fabric y that you buy.
b. Make an input-output table for the function for every 5 yards until you reach 30 yards
Section 1.7 TI-83 Activity – Creating a Table of Values
With this activity we will explore making tables of values with the TI-83 graphing calculator.
Step #1: Turn on the calculator.
Step #2: To create a table of values we will input a function.
Push the button in the upper left hand corner of the
calculator.
Step #3 Let’s work with the equation, y = 4x – 3
Input the right side of the equation into your calculator.
Step #4 When you are done entering the equation, push the (yellow)
button.
Step #5 Push the button on the top right hand side of the calculator.
Using the values the calculator computed, complete the following tables:
**If you can’t see the number for ‘x’ that you want, use the up and down arrow keys**
|X |Y |
|-3 | |
|-2 | |
|-1 | |
|0 | |
|1 | |
|2 | |
|3 | |
Step #6 After you have completed the table, go back to where you input your equation. Clear the current equation.
Step #7 Complete the given table of values for the following equation:
Y = 0.5x – 2
|X |Y |
|-3 | |
|-2 | |
|-1 | |
|0 | |
|1 | |
|2 | |
|3 | |
-----------------------
4.3 cm cmcm
7.2 cm
8 in.
6 in.
graph
2nd
Year PCs
’90 23.7
’92 32.4
’94 47.9
’96 70.8
’98 118.4
Group Number of
Species
Arachnids 5
Amphibians 9
Reptiles 14
Snails 15
Crustaceans 16
Insects 28
Mammals 59
Clams 61
Fishes 68
Birds 75
Y=
................
................
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