9TH Grade Mathematics [Algebra 1]
Required Textbook: A Beka Algebra 1, Work-Text, Pensacola, Florida, current edition.
--|Linear Expressions and Equations
--|Slope, Intercepts, and Slope-Intercept Form
--|Graphing Linear Equations
SYSTEMS OF EQUATIONS
--|Graphing Simultaneous Equations
--|Inconsistent and Dependent Systems
--|Monomials and Polynomials
--|Adding and Subtracting Polynomials
--|Prime, Composite, and Square Numbers
--|Factoring the Difference of Two Squares
--|Quadratic Expressions, Equations, And Functions
--|Solving Quadratic Equations by Factoring
--|Solving Quadratic Equations by Graphing
--|The Quadratic Formula
[a] Understand the basics of functions and know how to apply them to problem-solving situations.
[i] Identify the mathematical domains and ranges and determine reasonable domain and range values for given situations.
[ii] Know how to manipulate symbols in order to solve problems and use the necessary algebraic skills required to simplify algebraic expressions (factoring, properties of exponents etc.).
[ii] Connect the function notation of y = and ƒ(x) =.
[ii] Analyze situations and formulate systems of equations or inequalities in two or more unknowns to solve problems.
[iii] Interpret and determine the reasonableness of solutions to systems of equations or inequalities.
[iv] Identify and sketch graphs of parent functions, including linear (y = x), quadratic (y = x2), square root (y = √x), inverse (y = 1/x), exponential (y = ax), and logarithmic (y = logax) functions.
[b] More advanced functions.
[i] Describe a conic section as the intersection of a plane and a cone.
[ii] Identify symmetries from graphs of conic sections and identify the conic section from a given equation.
[iii] Understand that quadratic functions can be represented in different ways and learn to translate among their various representations.
[iv] Determine the reasonable domain and range values of quadratic functions, interpret and determine the reasonableness of solutions to quadratic equations and inequalities.
[v] Relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions.
[iv] Determine a quadratic function from its roots or a graph.
[v] Use characteristics of the quadratic parent function to sketch the related graphs and connect between the y = ax2 + bx + c and the y = a(x - h)2 + k symbolic representations.
[c] Rational functions.
[i] Formulate equations and inequalities based on rational functions.
[ii] Use quotients to describe the graphs of rational functions, describe limitations on the domains and ranges, and examine asymptotic behavior.
[iii] Determine the reasonable domain and range values of rational functions and determine the reasonableness of solutions rational equations and inequalities.
[iv] Analyze a situation modeled by a rational function, formulate an equation or inequality.
[d] Exponential and logarithmic functions.
[i] Formulate equations and inequalities based on exponential and logarithmic functions.
[ii] Develop the definition of logarithms by exploring and describing the relationship between exponential functions and their inverses.
[iii] Analyze a situation modeled by an exponential function, formulate an equation or inequality.
1.1 - Linear Expressions and Equations
Sometimes, you find yourself having to write out your own algebraic
expression based on the wording of a problem.
In that situation, you want to
The following are some key words and phrases and
read the problem carefully,
pick out key words and phrases and determine their equivalent mathematical
replace any unknowns with a variable, and
put it all together in an algebraic expression.
|Addition: sum, plus, add to, more than,
increased by, total
|Subtraction: difference of, minus,
subtracted from, less than, decreased by, less
|Multiplication: product, times, multiply,
|Division: quotient divide, into,
|[1.1.a] Write the phrase as an algebraic expression.
The sum of a number and 10
It looks like the only reference to a mathematical operation is the
word sum - so what operation will we have in this expression?
If you said addition, you are correct.
The phrase 'a number' indicates that it is an unknown number - there
was no specific value given to it. So we will replace the phrase
'a number' with the variable x. We want
to let our variable represent any number that is unknown
Putting everything together, we can translate the given english phrase
with the following algebraic expression:
The product of 5 and a number
*'sum' = +
*'a number' = variable X
*'The sum of a number and 10' = X + 10
Again, we are wanting to rewrite this as an algebraic expression, not
This time, the phrase that correlates with our operation is 'product'
- so what operation will we be doing this time? If you said
you are right on.
Again, we have the phrase 'a number', which again is going to be replaced
with a variable, since we do not know what the number is.
Let’s see what we get for this answer:
3 less than twice a number
*'product' = multiplication
*'a number' = variable X
The product of 5 and a number = 5X
First of all, we have the phrase 'less than' which mathematically
translates as subtraction. You need to be careful with this phrase,
it is very tempting to start off with 3 and put your subtraction sign after
the 3. However, think about it, if you want 3 less than something,
you are 3 below it. In order to be 3 below something, you would have to
subtract the 3. So you would not have 3 minus, but minus 3 as PART
of your expression.
The other part of the expression involves the phrase 'twice a number'.
'Twice' translates as two times a number and, as above, we will replace
the phrase 'a number' with our variable x.
Putting this together we get:
The quotient of 3 and the difference of a number and 2
*'less than' = -
*'twice' = 2 times
*'a number' = variable X
*'3 less than twice a number' = 3 - 2X
First of all, the term 'quotient' is going to be replaced with
what mathematical operation? If you said division, you are
right on the mark.
Note how 3 immediately follows the phrase 'the quotient of', this means
that 3 is going to be in the numerator. The phrase that immediately
follows the word 'quotient' is going to be in the numerator of it.
After the word ‘and', you have the phrase 'the difference of a number
and 2'. That is the second part of your quotient which means it will
go in the denominator. And what operation will we have when we do
write that difference down below? I hope you said subtraction.
Let’s see what we get when we put all of this together:
y = mx + b
*'quotient' = division
*'difference' = -
*'a number' = variable X
*'The quotient of 3 and the difference of a number and 2' = 3 / [X - 2]
1.2 - Slope, Intercepts, and Slope-Intercept Form
Straight-line equations, or linear equations, graph as straight lines, and have simple variables with no exponents on them.
For example, the linear function, f(x) = mx + b, may be graphed on the x, y plane as the equation,
This equation is called the slope-intercept form for a line.
The graph of this equation is a straight line.
The slope of the line is m.
The line crosses the y-axis at b.
The point where the line crosses the y-axis is called the y-intercept.
The x, y coordinates for the y-intercept are (0, b)
Given some information about a line, you can figure out the equation of that line.
[1.2.a]Find the equation of the straight line that has slope m = 4 and passes through the point (–1, –6).
First you write the standard equation of a line,
y = mx + b
Then plug in the numbers you are given. You've been given the slope m.
The point (-1, -6), gives me an x-value and a y-value: x = –1 and y = –6.
The slope-intercept form of a straight line has y, m, x, and b.
So the only missing component is is b (the y-intercept).
y = mx + b
y = 4x + b <-- replace m with 4
y = 4(-1) + b <-- replace x with -1
-6 = 4(-1) + b <-- replace y with -6
-6 = -4 + b
-6-(-4) = b
-2 = b
The line equation is y = 4x – 2
1.3 - Graphing Linear Equations
Follow the example y = 3x + 2 through this explanation.
The number in front of x is the slope.
(If necessary, place this number over 1 to
form a fraction.
This fraction is your rise/run.) slope = 3/1
The "b" value is where the line crosses the
y-axis. Be sure to check the sign of this
b = 2
Plot the b value on the y-axis.
see graph below
Standing at that point use your rise and run
values to plot your second point.
(If rise is positive, move up. If rise is
negative, move down.)
(If run is positive, move right. If run is
negative, move left.)
Connect the two points to form the line.
2.1 - Simultaneous Equations
When in a given problem, more than one algebraic equation is true at a time,
it is said there is a system of simultaneous equations which are all true together at once.
Such sets of multiple equations may help solve for more than one unknown variable in a problem,
since having more than one unknown in one equation is typically not enough information to solve any of the unknowns.
3.1 - Monomials and Polynomials
An algebraic expression consisting of one term that is a constant, a variable,
or the product of constants and variables is called a monomial. Examples of monomials are,
4 x 4y 7y2x3z
Two or more terms containing the same variables, with correspomnding variables having the same exponents
are called like terms. Examples of like terms are,
4x and x 5y and -7y
2x2 and 3x2 5z3 and 3z3
The degree of a monomial in one variable is the exponent of the variable. For example,
The degree of 4x is 1
The degree of 2x2 is 2
The degree of 5z3 is 3
The degree of a nonzero constant is zero. For example, 5 is the same as 5x0.
A monomial such as 7x3 may be considered a polynomial of one term.
A polynomial is on its simplest form when it contains no like terms. For example,
5z3 + 2x2 + 3z3 + 4
The terms 5z3 and 3z3 are like terms and can be added up.
The simplified form of the polinomial is,
8z3 + 2x2 + 4
4.1 - Prime, Composite, and Square Numbers
5.1 - Quadratic Expressions, Equations, And Functions
 A Beka Algebra 1, Work-Text, Pensacola, Florida.