Garden City
Public Schools
Algebra 1
Regents Review
Session One
1
Algebra
Session One

Real Number, Properties, Operations Review

Radicals- Simplifying

Expression/Equations/Inequalities

Geometry and Measurement

Sets/Relations/Functions/Graphing

Factoring
Session Two

Monomials/Polynomials/Law of Exponents

Rational Algebraic Expressions

Quadratic Equations

Coordinate Geometry
Session Three

Statistics- Standard Deviation, Box-and-Whisker, Regression Equations, Residuals

Ratio and Proportion

Two –Way Tables

Sequences and Series

Piecewise Graphs
2
Session One

Real Number, Properties, Operations Review

Radicals

Expression/Equations/Inequalities

Geometry and Measurement

Sets/Relations/Functions/Graphing

Factoring
3
Real Numbers, Properties and Operations Review
1. The set of natural numbers is the set of counting numbers.
2. The set of whole numbers is the set of all the natural numbers and zero.
3. The set of integers is the set of whole numbers and their opposites.
4. The set of rational numbers is the set of all numbers that can expressed as fractions in the form
a
where a and b are integers and b does not equal zero.
b
5. The set of irrational numbers is the set of all non-repeating, non-terminating decimals.
6. The set of real numbers is the set of all rational and irrational numbers. An irrational number
cannot be expressed as an integer or the quotient of integers.
7. The absolute value of a number is its measured distance from zero on a number line
8. Scientific notation is used to represent very large or very small numbers.
9. A number written in scientific notation is the product of two factors — a decimal greater than or
equal to 1 but less than 10, and a power of 10 (3.1
105= 310,000). Small numbers negative
exponents, large numbers positive exponents
10. A power of a number represents repeated multiplication of the number. For example, (–5)3
means (–5) · (–5) · (–5). The base is the number that is multiplied, and the exponent
represents the number of times the base is used as a factor. In this example, (–5) is the base,
and 3 is the exponent.
11. Any real number raised to the zero power is 1. The only exception to this rule is zero itself.
12. A perfect square is a whole number whose square root is a whole number ( 25 is 5; thus, 25 is
a perfect square).
13. The square root of a number is that number which when multiplied by itself equals the number.
14. Any whole number other than a perfect square has a square root that lies between two
consecutive whole numbers.
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15. The square root of a whole number that is not a perfect square is an irrational number ( 2 is an
irrational number). An irrational number cannot be expressed exactly as a ratio.
16. A set is said to be closed under a binary operation when every pair of elements from the set,
under the given operation, yields and element from that set.
Properties of Real Numbers
Regents Examples
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5
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25.
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Radicals
1. To simplify a radical that is not a perfect square:

find the factors of the radicand, one of which should be a perfect square

find the square root of this factor, leaving the remaining factor under the radical sign

Continue until there are no more perfect squares
Regents Examples
1.
10
2.
3.
4.
What is 3 √250 expressed in simplest radical form?
1) 5 √10
2) 8√ 10
3) 15 √10
4) 75 √10
5.
6.
7.
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Expressions, Equations and Inequalities
1. Algebraic expressions are evaluated by replacing the variables with numbers and applying the
order of operations to simplify the resulting expression.
2. An algebraic equation is a mathematical statement that says that two expressions are equal
3. Key words in translating verbal expressions/ sentences to algebraic expressions/equations may
include words and their translations such as: is to =, of to multiplication, more than to +, less than
to –, increased by to +, and decreased by to –.
4. An inequality is a mathematical sentence that states that one quantity is less than (or greater
than) another quantity.
5. To maintain equality, an operation that is performed on one side of an equation must be
performed on the other side.
6. When both expressions of an inequality are multiplied or divided by a negative number, the
inequality sign reverses (e.g., –3x < 15 is equivalent to x > –5)
7. An equation involving two or more variables is called a literal equation. In this type of equation,
we want to solve in terms of one of the other variable
7. Word Problems: In solving word problems, define each unknown (variable) in a ―Let‖ statement or a table.
Express as many of the unknowns in terms of your variable.

Consecutive Integers are integers that follow one another in order.
Example: {1, 2, 3, 4,…},{-5, -4, -3, -2,…}{ x, x + 1, x + 2, x + 3,…}
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
Consecutive even integers are even numbers that follow one another in order.
Example: {2, 4, 6, 8,…}, {-6, -4, -2, …}, {x, x + 2, x + 4,…}

Consecutive odd integers are odd numbers that follow one another in order.
Example: {1, 3, 5, 9,…}, {-5, -3, -1, …},{ x, x + 2, x + 4,…}

Coin Problems:
In solving problems which deal with coins of different value (ex. nickels dimes and quarters) it is
often helpful to represent the value of the coins in the same unit of money.
In the following example the unit of money is cents:
3 nickels is 15 cents (3 x 5)
2 dimes is 20 cents (2 x 10)
5 quarters is 125 cents (5 x 25)
$1.45 is 145 cents
In solving these problems you can organize your problem with a ―Let…‖ statement or a Table.
Ex. A boy has $3.20 in his bank made up of nickels, dimes and quarters. There are 3 times as many
quarters as nickels, and 5 more dimes than nickels. How many coins of each kind are there?
Let…
n = # of nickels
3n = # of quarters
(value of nickels in cents is 5n)
(value of quarters in cents is 25(3n))
n + 5 = # of dimes
(value of dimes in cents is 10(n + 5))
The total value in cents of all the coins is 320 cents.
Equation: 5n + 25(3n) + 10(n + 5) = 320

Distance = Rate x Time D = R x T

Age Problems
Regents Examples
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Geometry and Measurement
1. The perimeter of a polygon is the distance around the figure. In a circle this is the circumference
C
d or C 2 r
2. The area of a rectangle is computed by multiplying the length times width
3. The area of a triangle is computed by multiplying the measure of its base by the measure of its
height and dividing that product by 2.
4. The area of a parallelogram is computed by multiplying the measure of its base by the measure
of its height.
5. The area of a trapezoid is computed by taking the average of the measures of the two bases and
multiplying this average by the height.
6.The area of a circle is
r 2 (use
A
1
h(b1
2
A
b2 )
button on calculator)
7. The volume of a rectangular prism is V = lwh
8. The volume of a cylinder is computed by multiplying the area of the base, B, ( r2) by the height
of the cylinder V = r2h.
9. Surface Area: rectangular prism
cylinder
SA
SA 2lw 2hw 2lh
2 r2
2 rh
10. Calculate the relative error in measuring square and cubic units, when there is an error in the
linear measure with the following formula
Percent of error
measured value true value
true value
100%
Regents Examples
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27
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Sets
 A set is a collection of objects or elements, such as a set of numbers. The universal set is the
set of all elements in a given situation.
 The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that belong
to both sets, A and B. Two sets are disjoint if there intersection is the empty set
 The union of two sets, A and B, denoted A U B, is the set of all elements that belong to set A
or set B, or to both set A and set B.
 The complement of a set A, denoted A , is the set of all elements that belong to the universe,
U but do not belong to set A
 Inclusive set notation [
] includes endpoint; exclusive ( ) does not include endpoint.
 Set builder notation is a mathematically concise way of describing a set without listing the
elements of the set. For example, using the set-builder notation, the set of counting
numbers from 1 to 100 is:
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{x | x is a whole number and 1≤ x ≤100}
This reads as “the set of all x such that x is an integer and x is greater than or equal to 1 and less
than or equal to 100.” The vertical bar ―|‖ represents the phrase ―such that,‖ and the description to
the right of the bar is the rule that defines the set. This replaces listing the numbers as {1, 2,… 97,
98, 99, 100} Frequently used with set-builder notation is the symbol ― ‖ which means ―is an
element of‖ and the symbol ― ‖ which means ―is not an element of‖.
Let’s look at the following example…. The diagram below shows two sets of numbers, set A and set B.
1. List the elements of set A
2. List the elements of set B
3. List the elements of the Universal set, U
4. List the elements of A
B
5. List the elements of A
B
6. List the elements of A
7. List the elements of B
Relations and Functions
1. A relation is any set of ordered pairs
2. The domain of a relation is the first element of the set of ordered pairs
3. The range of a relation is the second element of the set of ordered pairs
4. A function is a special type of relation in that each element of the domain corresponds to one and
only one element in the range.
5. Graphs of functions take on many different forms: linear, quadratic, absolute value and
exponential, both growth and decay.
6. The vertical line test for the graph of a relation insures that it is a function.
7. An exponential function is a function of the form, y
b x , where b is positive and the variable is
contained in the exponent.
8. When b > 1, then the exponential function increases; when 0 < b < 1, then the exponential function
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decreases.
9. Growth and decay problems are shown by the following formula: y
A(1 r ) x , where A is the
initial value, r is the rate of growth ( + ) or decay ( − ) and x is time period given in the problem.
Regents Examples
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1.
FactoringThe GCF is the largest factor of a set of numbers. When you are given a set of
numbers, write all the factors of that number. Compare this list which each number. Circle
largest common number.
2. When factoring trinomials into binomial factors place two parentheses under trinomial before
factoring, then:
a. Find the square root of the first term of the polynomial, usually x.
b. Find factors of the last term of the polynomial that sum to the middle term
c. When the sign in front of the third term is ―+‖, the binomials will have the same sign,
either both positive or both negative; the sign in front of the second term will decide
which sign.
d. When the sign in front of the third term is ―−‖, the binomials will have different signs, and
the sign in front of the second term will be the sign of the larger factor.
e. When factoring the difference of two squares (DOTS): place parentheses under the
expression then find the square root of each term one factor will have a + sign; the other
−.
3. To factor a polynomial completely identify and factor the GCF of the polynomial followed by the
factoring of the trinomial (if applicable)
Regents Examples
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