Experimenting with Lines

Constructing Algebra

By Germán A. Moreno

1.1 Number Sets and the Structure of Algebra

Example 1

x / C

Example 2

Example 3

Example 4

Example 5

Set Theory

Cantor described a set as a collection of definite, distinguishable objects of our perception to be conceived as a whole. The individual objects are called elements of the set. Sets are finite, infinite, or empty.

What do the words definite and distinguishable mean in every day language?

Example 6

The intersection of the set of all cars with the set of all trucks.

Example 7

Give an example of a finite, infinite and an empty set.

The set of vowels {a, e, i, o, u}

The set of years {…,-500 B.C.,…,0 A.D.,…500 A.D.}

Example 8

Example 9

Example 10

1.2 Fractions

Example 1

Example 2

{15, 30, 45, 60,…}

Example 3

Multiples of 14: 14, 28, 42, 56,

Multiples of 21: 21, 42,63, 84,

Example 4

Example 5

Example 6

Example 7

2 is a prime number.

Example 7

Example 8

1.3 Adding and Subtracting Real Numbers; Properties of Real Numbers

Example 1

Example 2

Example 3

Example 4

Example 5

Using the Properties and Theorems

Example 6

Add or subtract the following by using the properties and theorems.

1.4 Multiplying and Dividing Real Numbers; Properties of Real Numbers

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Using the Properties and Theorems

Example 7

Evaluate.

1.5Order of Operations

Warm Up Activity 1

Complete the statement so that it is true.

1.)

2.)

3.)

4.)

5.)

Example 1

Example 2

The Order of Operations is very important when simplifying expressions and equations. The Order of Operations is a standard that defines the order in which you should simplify different operations such as addition, subtraction, multiplication and division.

This standard is critical to simplifying and solving different algebra problems. Without it, two different people may interpret an equation or expression in different ways and come up with different answers. The Order of Operations is shown below.

Example 2

Example 3

Example 4

Example 5

Application Problems

Example 6

A teacher calculates his students’ grade as follows. Sixty percent of the final grade comes from their test average and 40 percent comes from their homework average. Roberto’s homework average is a 90 and his 4 test grades were 88, 89, 90, and 87. What will be his final grade?

1.7 Combining Like Terms

Example 1

Example 2

Evaluate Expressions

We have learned that, in an algebraic expression, letters can stand for numbers. When we substitute a specific value for each variable, and then perform the operations, it's called evaluating the expression.

Example 3

Evaluate when y = 5.

Example 4

Evaluate when u = 2 and v = 7.

Example 3

Combining Like Terms

When we talk about combining like terms we use the operations addition and subtraction.

Rule Number 1: With all operations you treat the coefficients as you would if they were just part of a problem that did not have variables.

Rule number 2: When you add or subtract like terms the power of the variable does not change. The only thing that changes is the coefficient like stated in rule 1.

Example 4

Example 5

Example 6

Example 7

2.1 Equations, Formulas, and the Problem Solving Process

How to Solve It

UNDERSTANDING THE PROBLEM
First. You have to understand the problem.
What is the unknown? What are the data? What is the condition?
Draw a figure. Introduce suitable notation.
DEVISING A PLAN
Second. Find the connection between the data and the unknown. You should obtain eventually a plan of the solution.
Have you seen it before? Or have you seen the same problem in a slightly different form?
Do you know a related problem? Do you know a theorem that could be useful?
Could you restate the problem? Could you restate it still differently? Go back to definitions.
If you cannot solve the proposed problem try to solve first some related problem. Could you solve a part of the problem?? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown?
Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?
CARRYING OUT THE PLAN
Third.Carry out your plan.
Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?
Looking Back
Fourth.Examine the solution obtained.
Can you check the result? Can you check the argument?
Can you derive the solution differently? Can you see it at a glance?

2.2 The Addition Principle

Example 1

Example 2

Example 3

Example 4

Example 5

Application Problems

Example 7

Robert knows the distance from his home to work is 42 miles. Unfortunately he gets a flat tire 16 miles from work. How far did Robert drive before his flat tire?

Example 8

The perimeter of the trapezoid shown is 100 yards. Find the length of the missing side.

Example 9

In a survey respondents were given a statement and they could agree, disagree, or have no opinion regarding the statement. The results indicate of the respondents agree with the statement while disagree. What fraction had no opinion?

2.3 The Multiplication Principle

Example 1

Example 2

Example 3

Example 4

Example 5

Application Problems

Example 5

Example 6

Example 7

2.5 Translating Word Sentences to Equations

Keywords that involve Addition

sum
total
more than
greater than
consecutive
increased by
plus
older than
farther than

Example 1:

The sum of the length and width is 30.
The length is 4 more than the width.
The length is greater than the width by 5.
The consecutive integer after the integer n is 20.
Jane is two years older thanAlice.
Alice ran 5 kilometers farther than Jane.

Keywords that involve Subtraction

difference
diminished by
fewer than
less than
decreased by
minus
subtracted from
younger than

Example 2:

The difference between Jane’s age and Alice’s age is 10.
John has 10 fewer coins thanAlice.

Keywords that involve Multiplication

product
twice
doubled
tripled
times
multiplied by
of

Example 3:

The product of the length and the width is 50.
The length is twice the width.
John took half the number of marbles.

Keywords that involve Division

quotient
divided by
divided into
quotient of
in
per

Example 4:

The length divided by the width is 5.
John traveled 100 miles in 2 hours.

Example 5

Eight times the sum of eight and a number is equal to one hundred sixty.

Example 6

The product of three and the sum of a number and four added to twice the number yilds negative three.

Example 7

Five times the sum of a number and one-third is equal to three times the number increased by two-thirds.

Example 8

Translate the equation to a word sentence.

Example 9

Translate the equation to a word sentence.

2.6 Solving Linear Inequalities

Example 1

Example 2

Graphing Inequailties:

1. If the symbol is or draw a [ or a on the number line on the indicated number. If the symbol is < or > draw ( or an on the number line on the indicated number.

2. If the variable is > the indicated number, shade to the right. If the variable is < the indicated number, shade to the right.

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

Application Problems

Example 9

The design of a storage box calls for a width of 27 inches and a length of 41 inches. If the surface area must be at least 4254 square inches, find the range of values for the height.

Example 10

In Aaron’s English course, the final grade is determined by the average of five papers. The department requires any student whose average falls below 75 to repeat the course. Aaron’s scores on the first four papers are 68, 78, 80 and 72. What range of scores on the fifth paper would cause him to have to repeat the course?

4.1 The Rectangular Coordinate System

Warm-Up Activity 2

Graph the following points.

(-5, -3.4)
(8.5,-3)
(-6.4,6)
(2/3,4/3)

Making Scatter Plots of Data

Make a different scatter plot for each of the following data sets.

x / y
3 / 8
5.5 / 7.1
8 / 6.2
12 / 5
16 / 4.4
18.5 / 3.4
23 / 2
25.5 / 1
30 / 0.3
31 / -1

x / y
-5 / 10
-15 / 20
-20 / 30
-25 / 40
-30 / 50
-40 / 60
-50 / 70

x / y
-6 / 5
-2.5 / 8
-.5 / 12
0.9 / 15
1.8 / 18
3.5 / 22
5.9 / 25
8 / 30

x / y
-2 / 26
-1 / 17
0 / 10
1 / 5
2 / 2
3 / 1
4 / 2
5 / 5
6 / 10
7 / 17
8 / 26

5. Choose points that lie on a line and then make a scatter plot of the data.

x / y

6. Choose points that make a pentagon then makea scatter plot of the data.

x / y

4.7 Introduction to Relations and Functions

Example 1

Example 2

Construct any relation with the specified number of elements.

4 elements

5 elements

6 elements

Example 3

Identify the domain and range of each relation from Example 2

Example 4

The Vertical Line Test

The vertical line test is a way to determine whether or not a relation is a function. The vertical line test simply states that if a vertical line intersects the relation's graph in more than one place, then the relation is a not a function.

Example 5

Activity I

Relations and Functions

Determine whether or not the following relations are functions. Explain how you know that it is or is not a function.

X / Y
1 / 1
3 / 3
5 / 4
7 / 7
9 / 9
11 / 0
13 / 8
15 / 9

X / Y
-10 / -8
-6 / -3
-4 / 0
-1 / 1
0 / 3
1 / -2
2 / -4

X / Y
1 / 1
3 / 3
1 / 4
7 / 7
9 / 9
3 / 0
13 / 8
15 / 9

X / -3 / 5 / 2 / 9 / -2 / 2 / -7 / 6 / 0
Y / 1 / 6 / -9 / -5 / 4 / 3 / 6 / -7 / 0

10.

Activity II

Identify the Domain and Range

5. Draw a line with the following domain and range.

6. Draw any curve with the following domain and range.

4.2 Graphing Linear Equations

Example 1

(3,-5)

(6,3)

(1,-9)

To find a solution to an equation in two variables:

Choose a value for one of the variables.
Replace the corresponding variable with your chosen value.
Solve the equation for the value of the other variable.

Example 2

To graph a linear equation:

Find at least two solutions to the equation.
Plot the solutions as points in the rectangular coordinate system.
Connect the points to form a straight line.

Example 3

Each of the following is a function. Choose input numbers as indicated and fill out the table. Then graph the data.

Choose any 5 real numbers as inputs. Make sure they are not all whole numbers and not all positives.

Choose any 5 positive integers as inputs.

Choose any 5 integers as inputs. Choose positives and negatives.

Example 4

Example 5

Horizontal Lines

Example 6

Vertical Lines

Example 7

Application Problems

Example 8

Example 9

4.3 Graphing Using Intercepts

Example 1

Finding the x-intercepts

To find an x intercept.

Replace y with 0 in the given equation.
Solve for x

Finding the y-intercepts.

To find a y-intercept

Replace x with 0 in the given equation.
Solve for y.

Example 2

Equations in standard form.

Example 3

Equations in slope-intercept form.

Example 4

Equations in y = mx form.

Example 5

Equations in y = c form.

Example 6

Equations in x = c form.

Application Problem

Example 7

Slope-Intercept Form.

Example 1

Sally and her mom are riding bikes to the park. The graphs below show the distance traveled over time. Who is traveling faster?

Example 2

Experimenting with lines

Slope

In your calculator or on a graph, graph any line with a positive slope. If you are using a calculator sketch the graph on you paper.

Choose a different number for the slope (choose a positive number) and graph the new line on the calculator or on the same graph. If you are using a calculator sketch the graph on you paper.

Choose a different number for the slope (choose a positive number) and graph the new line on the calculator or on the same graph. If you are using a calculator sketch the graph on you paper.

Follow steps 1-3, but now use negative numbers for all the slopes. When you are done with this you should have 6 different lines.

6. In a few sentences explain the effect of changing the value for slope on the line.

Y-intercept

In your calculator or on a graph, graph any line. If you are using a calculator sketch the graph on you paper.

Choose a different number for the the y-intercept and graph the new line on the calculator or on the same graph. If you are using a calculator sketch the graph on you paper.

Choose a different number for the y-intercept and graph the new line on the calculator or on the same graph. If you are using a calculator sketch the graph on you paper.

In a few sentences explain the effect of changing the value for y intercept on the line.

Example 3

Compute the slope of the line that goes through the given points. Then plot the points and the line that goes through them on a graph.

1.(1, 2), (6, 5) 2. (1, 3), (2,-1)

3. (-5,-6), (10, 11) 4. (-1, 6), (3, 15)

Example 4

The graph shows how much gasoline will be used on the family vacation this year. Determine the slope of the line.

Example 5

After spending a day at a water park, Sue returned her inner tube with mud on it and now she must pay a cleaning fee. The solid line represents the cost of renting an inner tube. The dashed line represents the rental cost plus the cleaning fee. What is the slope of each line? How much is the cleaning fee?

Graphing Equations in Slope-Intercept Form

Example 4

Determine the equation of a line given two points on the line.

Example 5

Pattern Block Activity

Algebra can be used to describe patterns. In this activity you will write linear equations that model a pattern. For each of the pictures below you must suppose that the object is lying on the floor and that you are being asked to determine how many faces of the object are exposed. For each of the pages:

Fill in the table shown.

Determine the function rule for the pattern that expresses how many faces are exposed.

Make a scatter plot and graph the function rule in the calculator. Sketch the graph of the scatter plot and the function rule.

N / Picture / Explanation / Process / F(n)
1 /
2 /
3 /
4 /
10
15
N
N / Picture / Explanation / Process / F(n)
1 /
2 /
3 /
4 /
10
15
N
N / Picture / Explanation / Process / F(n)
1 /
2 /
3 /
4
10
15
N
n / Picture / Explanation / Process / F(n)
1 /
2 /
3 /
4 /
10
15
N

Point Slope Form

Point-Slope form

Example 1

Standard Form

Example 2

Write the equation of the line represented by the table and graph in standard form.

x / y
-3 / 1.1
-2 / 1.4
-1 / 1.7
0 / 2
1 / 2.3
2 / 2.6
3 / 2.9

Example 3

Write the equation of the line that passes through (2,3) and is parallel to the line below.

x / y
-3 / -5
-2 / -3
-1 / -1
0 / 1
1 / 3
2 / 5
3 / 7

Example 4

Write the equation of the line that passes through (2,3) and is perpendicular to the line below.

x / y
-3 / -5
-2 / -3
-1 / -1
0 / 1
1 / 3
2 / 5
3 / 7

4.6Graphing Linear Inequalities

Graphing Linear Inequalities in Two Variables

To graph a linear inequality in two variables (say, x and y), first get y alone on one side. Then consider the related equation obtained by changing the inequality sign to an equals sign. The graph of this equation is a line.

If the inequality is strict ( or ), graph a dashed line. If the inequality is not strict ( or ), graph a solid line.

Finally, pick one point not on the line ((0, 0) is usually the easiest) and decide whether these coordinates satisfy the inequality or not. If they do, shade the half-plane containing that point. If they don't, shade the other half-plane.

Example 1

Example 2

Application Problems

Example 3

Pg. 334 #41

Example 4

Pg. #42

8.1 Solving Systems of Linear Equations

Example 1

Problem 1

The school’s rocket club wants to travel to WashingtonD.C. for a competition. They have built a 4 ft. rocket for the competition and they think they can send it up 2000 ft in the air. The club must hold a fund raiser to raise the money. They decide to sell chocolates door to door to raise money, but they have found two companies that sell the chocolates. Chocolates-Are-Us charges a $50 base fee and then charges $0.90 per chocolate. Chocolates-Chocolates charges an $80 base fee and then charges $0.81 per chocolate. Answer the following questions.

Fill out the table below to determine which is the better offer.

Number of chocolates / Process column (Chocolates-Are-Us) / Cost at Chocolates-Are-Us / Process column (Chocolates-Chocolates) / Cost at Chocolates-Chocolates
10
20
30
60
90
120
240
480
N

2. Write a function rule for the cost of chocolates from Chocolates-Are-Us.

3. Write a function rule for the cost of chocolates from Chocolates-Chocolates.

4. How many chocolates (approximately) can you buy from Chocolates-Are-Us for

$190? $310?

How many chocolates (approximately) can you buy from Chocolates-Chocolates for $190? $310?

6. Graph both deals on the same graph below.

Determine at which point the cost is the same for the same amount of chocolates (approximately) and write down the number of chocolates and the cost below.

Example 2

On Your Own

Problem 2

The school choir would like to travel to a competition in from El Paso to WashingtonD.C. The choir must hold a fund raiser to raise the money. They decide to sell candies at a school fair to raise money, but they have found two companies that sell the candies. Company X charges a $40 base fee and then charges $0.50 per candy. Company Y charges a $100 base fee and then charges $0.35 per candy. They will be selling the candies for $2.00.

Answer the following questions.

Fill out the table below to determine which is the better offer.

Number of roses / Process column (Company X) / Cost at company X / Process column (Company Y) / Cost at Company Y
10
20
30
60
90
120
240
480
N

2. Write a function rule for the cost of candies for Company X.

3. Write a function rule for the cost of candies from company Y.

4. How many candies can you buy from Company X for $200? $300? $350?

5. How many candies can you buy from Company Y for $200? $300? $350?

6. Graph both deals on the same graph below.

7. Determine at which point the cost is the same for the same amount of candies approximately and write down the number of candies and the cost below.

Walking Experiment

The objective of this experiment is to obtain data about you and your partners walking speed. Once you have collected raw data from your experiment, you are to determine the equation that models your walking.

Materials:

4 partners

timer

calculator

notepad

Procedure:

When you get to the track, practice walking at a constant rate. That means that you should not speed up or slow down as you walk.
Choose one of the partners to be the first walker.
Someone should be chosen to write down the data using the table provided below.
Begin timing him/her from the 0 yard line.
Another partner should walk along with the walker and notice the distance the walker has traveled.
Every thirty seconds the timer should request the distance of the walker from the partner mentioned in 4.
Data should be collected every thirty seconds so that there are at least 10 data points.
Repeat steps 1-7 for a different walker at a different speed.

x f(x)x f(x)