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Chapter One: From Patterns to Algebra
Section One: Using Differences to Identify Patterns
In this lesson we will be looking for patterns in a number sequence. Patterns occur not
only in math but everywhere in the real world. Three dots at the end of a number
sequence indicate that the list does not stop there. If the difference from one term to the
next is always the same, the difference is called a constant difference.
EX1: Find the next three numbers in the sequence by using constant differences.
a. 1, 4,7,10,13,...
b. 30, 25, 20,15,10,...
It’s not always the first difference that is a constant.
EX2: Find the next three numbers in the sequence by using constant differences.
a. 2,6,12, 20,30,...
b. 8, 20,30,38, 44,...
Even though we cannot be 100% sure that a scenario will continue following a certain
pattern, most of the time we assume that it probably will. We therefore are making a
conjecture about the statement. A conjecture is a statement about observations that is
believed to be true.
EX3: The table below shows the relationship between temperatures in Celsius and
temperatures in Fahrenheit.
Celcius
0 10 20 30 40
Fahrenheit 32 50 68 86 104
Extend the table to find the Fahrenheit temperature that corresponds with Celsius
temperatures of 50, 70, and 100.
EX4: Suppose that 10 friends have just returned to school. Each friend has exactly one
conversation with each of the other friends. Determine how many total conversations
there will be.
Section Two: Variables, Expressions, and Equations
Suppose that a bike shop rents out bikes. They charge an $8 fee plus
$3 per hour to rent a bike. Create a chart that tells how much it will
cost to rent the bike for different amounts of time.
Hours Cost
0
1
Variables are letters that take the place of unknown quantities in
2
mathematical statements. Since we are not sure how many hours that a
3
customer will be renting a bike, we can use a variable to take the place
4
of this unknown. The cost of renting the bike is $8 plus $3 for each
hour. This can be written as 8  3  h . This is called an algebraic expression. We could
now use this expression to find the cost of renting the bike for any amount of time. When
plugging different values into an expression, the process is called evaluating the
expression.
EX1: Evaluate the expression 8  3  h to find out how much it would cost to rent the bike
for 6, 8, and 12 hours.
An equation is a mathematical statement that says that two things are equal. (Notice that
the root word of equation is equate.) You need an equal sign for equations.
EX2: The bike shop also rents out in-line skates. They charge $2 per hour plus a $4 up
front fee. Write an expression that represents the cost of renting the skates. Then write an
equation that relates the expression to a total cost of $10.
The solutions to an equation is the value that you can substitute into the variable that
makes the statement true.
EX3: Plug values into the previous equation to find one that makes it true.
EX4: Jessica places 3 photos on the cover of her photo album. She places 4 photos on
each of the other pages. She has 155 photos in all. How many pages will she need if she
uses all of her photos?
EX5: Marge’s class is going on a field trip to the aquarium. The class rate is $12 per child
plus $25. How many students can go if the class has $335 to spend?
We also have a method of using a calculator to find the solution to some equations.
EX6: In 1990, the United Nations reported a deforestation figure of around 114,364 acres
per day. The area of Olympic National Park is about 922,654 acres. How long would it
take to lose an area of forest equal to the area of Olympic National Park?
Section Three: The Algebraic Order of Operations
When simplifying algebraic expressions we must follow an order of operations: Please
Excuse My Dear Aunt Sally or PEMDAS.
Steps of PEMDAS
1. Parenthesis (This includes all grouping symbols ( ),   ,   ,
2. Exponents
3. Multiply and Divide (Do these in order from left to right)
4. Add and Subtract (Do these in order from left to right)
)
EX1: Use order of operations to simplify each expression.
a. 13  2  5
b. 62  4  3  7 2
c.  3  4  7    2  4  3  4
EX2: Insert grouping symbols to make 80  2  6  5  50 true.
When we evaluate expressions we must follow proper order of operation as well
EX3: If a  2, b  4, and c  5 , evaluate the following expressions
a. a 2  b 2
b. a2   b  c 
c. b 2   c 2  a 
Sometimes we have implied parenthesis when evaluating expressions
EX4: Use a graphing calculator to evaluate the expression
17  97 42  6

2
12
Parentheses are evaluated from the inside out.


EX5: Evaluate the following expression 3 2 2  3  1  1  1
Section Four: Graphing with Coordinates
When graphing in mathematics we use a coordinate plane (sometimes called a Cartesian
plane). It is made up of a horizontal number line (x-axis) and a vertical number line (yaxis). The intersection of the axes is called the origin. The two number lines divide the
plane into 4 equal parts called quadrants. Any coordinate can be plotted on the graph. The
coordinates, or location, of a point on the coordinate plane is called an ordered pair. The
coordinates of the origin is the ordered pair  0,0  .
EX1: Determine the quadrant and coordinates for all the points on the graph.
EX2: Graph the points K  3,8 , L  5, 7  , M 8, 2 , N  0, 3 , and P  5,0 
EX3: Make a table for the following equations and find values of y by substituting 1, 2, 3,
4, and 5 for x. Graph the ordered pairs on a coordinate plane, and connect the points to
make a line.
a. y  3x
b. y  2 x  1
c. y  3x  2
Section Five: Representing Linear Patterns
A DVD buyers club charges a $20 membership fee and $15 per DVD purchased. Finish
the table that represents this situation.
DVDs
0
1
2
3
4
5
Cost
A constant difference in both x’s and y’s tell us that we have a linear relationship. This
means that if we graph the ordered pairs, it creates a straight line. What is an equation
that represents the scenario?
EX1: The table shows the costs associated with an audio CD club that charges a
membership fee. Write an equation to represent the pattern in the data.
CDs 0 1 2 3 4 5
Cost 21 31 41 51 61 71
When an equation is graphed and the resulting graph is a straight line, we call the
equation a linear equation. In every linear equation, there is an independent variable and a
dependent variable. The independent variable (usually x) is the variable that is normally
on the left of the t-table and that you plug into the equation to find the dependent variable
(usually y). Therefore, the dependent variable is so called because it depends on the other
variable. On the graph the independent variable is on the horizontal axis and the
dependent is on the vertical axis.
EX2: A passenger train averages 54 mph. Write a linear equation that represents the
distance traveled in a given time. Graph the equation for times from 0 to 15 hours.
EX3: Exercise physiologists suggest that a reasonable estimate for the maximum heart
rate during exercise is no more than 220 beats per minute minus the person’s age. Write a
linear equation to express the maximum heart rate in terms of age. Graph the equation for
ages 10 to 50. What are reasonable values for the independent variables?
Section Six: Scatter Plots and Lines of Best Fit
Not always will real life data lie in a perfect line. However many times the data, even
though not perfect, will follow closely to a pattern. We may see a trend in the data. An
effective way to look for these trends is by using a scatter plot. A scatter plot is simply a
cluster of points. Sometimes the cluster will closely resemble a line. If there seems to be a
relationship such as a line, we say that we have a correlation. There are two types of
correlations that resemble a line. A positive correlation (slants up) and a negative
correlation (slants down).
In a positive correlation as the x-values increase, the y-values do the same. In a negative
correlation as the x-values increase, the y-values decrease.
Positive Correlations
 Age and height
 Education and income
 Hours of working out and strength
Negative Correlations
 Hours spent on sports and hours of free time
 Days of running and minutes it takes to run a mile
 Hours of sleep and number of times your head nods in class
No Correlations
 Age and eye color
 Money you possess and number of siblings you have
If we have a correlation, then we should be able to closely fit a straight line through the
points. This line is called the line-of-best-fit. This line can be used to make predictions
about the data.
EX1: Make a scatter plot for the data below on the number of people working on farms in
various years, and draw the line of best fit. Describe the correlation as strong positive,
strong negative, or little to none.
Year
1940 1950 1960 1970 1980 1990
Number of
farm workers 8995 6858 4132 2881 2818 2864
in thousands
Use the line of best fit to predict the number of number of farm workers in the year 2000.