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S
T E P: Science, Technology, Engineering, and Mathematics Talent Expansion
Program
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Dream
Catchers Project
 Basic Mathematics;
Basic Algebra |
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SPEAK THE LANGUAGE OF
DREAMERS |
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CLICK HERE
FOR WORKSHEETS |
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i. THE NUMBER LINE
The number line is a visual representation of the set
of Real Numbers. Zero is neither positive nor negative. It is helpful to make the distance
between the numbers (the scale) uniform.
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ii. GRAPHING NUMBERS ON THE NUMBER LINE
The number paired with a point on the number line is called the
coordinate of that point on the line. The point paired with a number is called the
graph of
that number.
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iii.
ORDER RELATION ON THE NUMBER LINE
The order of points on the number line is the order of the
numbers. Larger numbers are located to the right of smaller numbers on the
number line.
The mathematical symbols for order relation are as follows:
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iv. SETS AND SET NOTATION
A set is a collection of objects. The objects in a
set are called elements or members and are written between set
braces.
The set containing the whole numbers between 0 and 6 is
written as: {1,2,3,4,5}.
INFINITE SET: The set of all whole numbers has
an infinite number of elements. That set can be written as:
{0,1,2,3,4,5...}. The three dots indicate that the roster continues
without coming to an end and is called an infinite set.
FINITE SET: A finite set is a set in which the elements can be
counted.
OTHER WAYS TO DESCRIBE A SET: Using a list is not the only way to describe a set. It is not possible to list all the elements of the infinite set:
{all fractions between 0 and 1}. The finite set {all even numbers between
2 and 12} could also be described using the following list:
{4, 6, 8, 10}.
EMPTY SET OR NULL SET: A set can contain no elements and is called the
empty set
or the null set. The empty set is designated using braces without
any elements in between them { } or with a special symbol
Ø,
written without braces.
The set {0} is not the symbol for the empty set.
This set is the set containing 0 as its only element.
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v. USING SET NOTATION
Answers to math problems are often expressed as elements of a
set.
In the equation, x + 4 = 9, the answer for the x can be written
using set notation such as: x = {5} or {x | 5}.
The first answer tells the reader that the answer for x is 5. The second
answer uses set builder notation and reads as follows: the
values for x such that x is equal to 5. Say the words "such that" for the
vertical line in set builder notation.
In the equation, x² + 4 = 8, the values for x can be written
as x = { -2, 2 } or { x | -2, 2 }. The second answer in set builder
notation is read as follows: the values
for x such that x is equal to -2 or 2.
Set builder notation is helpful to describe infinite sets.
For example { x | x is a quotient of two integers, with denominator not 0}
describes the set of rational numbers. The information inside
the braces is read as "the set of numbers x such that x is the quotient of two
integers, with denominator not 0."
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vi. DOMAIN
A variable is a symbol that is used to represent values for any
element in a set. The domain is the set of numbers from which the values
for the variable can be chosen.
In the equation, x + 7 = 10, the value for x is selected from the
domain that contains all Real numbers. The domain
can be written as follows: { x | all Real
numbers }. The only value from the set of Real numbers that will satisfy
the given equation is x = { 3 }.
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vii. THE NEGATIVE SYMBOL & THE OPPOSITE
There are several names for the negative symbol.
Other names include minus, subtract and opposite. The negative
symbol is a separate symbol from the number. A negative sign in front of a
number means that number is found to the left of zero on the number line and is
called negative. Two negative signs in front of a number mean the opposite
of a negative.
Since the opposite of negative is positive, the opposite of a
negative number is a positive number.
– ( – 2 ) = + 2.
A positive number may or may not have a + sign in front.
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viii. ABSOLUTE VALUE
The symbol for absolute value is two parallel vertical bars
which contain a numerical or algebraic expressions.
| 2 | is read the absolute
value of 2.
| –2 | is read the absolute
value of negative 2.
Absolute value is the distance from
zero to the expression between the vertical bars on the number line. Since
distance is always positive, absolute value will always be positive.
| 2 | = 2 and |
– 2 | = 2
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ix. GROUPING SYMBOLS
Grouping symbols are used to indicate the order in which the
operations should be performed in a math problem. Grouping symbols
include: parenthesis ( ), brackets [ ], braces { }, and
the fraction bar.
6 + (3 × 5) indicates that the multiplication
is completed before the addition.
When grouping symbols are absent, operations are performed
according to the Order of Operations Agreement.
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xi. OPERATIONS WITH REAL NUMBERS
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xii.
ADDITION AND SUBTRACTION |
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xiii.
To add or subtract two numbers on the number line,
start at the position of the first number, then move a distance
equal to the second number in the direction associated with the sign of
the second number.
Move to the left if the second number is negative.
Move to the right if the second number is positive.
The number at the resulting position is the answer.
3 + 4 = 7
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3 + − 4 = − 1
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xvi. PROBLEMS THAT
LOOK LIKE SUBTRACTION
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Problems that look like subtraction are addition
problems with negative numbers.
Use the addition rules for problems that look like
subtraction..
6 – 9 = – 3 because 6 – 9 is the same as 6 + – 9
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xvii. MULTIPLICATION AND DIVISION
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Because division can be defined as multiplication by the
reciprocal, multiplication and division problems have the same rules. |
| Multiply and Divide two numbers with the same
signs: The answer is always positive.
6 • 9 = 54, – 6 • –9 = 54, and
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Multiply and Divide two numbers with
unlike signs:
The answer is always negative.
– 6
• 9 = – 54, 6 • – 9 = – 54, and
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i.
VARIABLE EXPRESSIONS
Expressions do not contain equals signs. Variable
expressions contain variables and numbers. The value of a variable
expression may change depending on the value chosen to replace the variable or
variables. The value for the variable is chosen from its
domain.
| Examples of variable expressions are: |
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| 4x + 5; |
–2c; |
7a
– b; |
⅓Y + 2; |
–
6m – 10n; |
a + b + c; |
8z |
ii. EVALUATING EXPRESSIONS
This is the process for finding the number equal to a
variable expression. To evaluate
an expression, select a value for the variable from the
domain, substitute the value for the variable, and complete the problem
according to the order of
operations agreement
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iii. SIMPLIFYING VARIABLE EXPRESSIONS
by COMBINING LIKE TERMS
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VOCABULARY |
EXPLANATION |
EXAMPLE: 3x² + 2xy – 5x + y – 8 +
9x² + 12 – 4y |
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Variable
Expressions |
contain terms
which are separated by plus or minus signs. The terms can be either numerical or variable. |
contains eight terms, both
numerical terms and variable terms |
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Numerical Terms
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contains
only numbers. |
Numerical terms are: – 8 and 12 |
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Variable Terms |
contain variables and have two
parts: 1. the variable part and 2. the numerical
coefficient. |
Variable terms are: 3x²,
2xy, –5x, y, 9x², and –4y |
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The Numerical Coefficient |
is the number in front of the variable.
Variable terms that appear to be only variables have a coefficient of
one. |
The coefficients of the variable terms in the
expression are: 3, 2, –5, 1,
9, and –4. |
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Like Terms
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are terms that are identical in the
variable part. Like terms are different from like signs.
Like terms have the same variables with the same exponents. |
There are three pairs of like terms:
(1) 3x²
and 9x²,
(2) y and –4y, and
(3) –8 and 12.
–5x is not like 3x² and 9x² because the exponent is
different.
–8 and 12 are like terms because they are both numerical
terms. |
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Combining Like
Terms |
Like terms may be combined
by adding their coefficients using the addition rules. |
3x² + 2xy – 5x + y – 8 +
9x² + 12 – 4y =
12x²
+ 2xy – 5x – 3y + 4. |
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x. SIMPLIFYING VARIABLE EXPRESSIONS by using
MULTIPLICATION AND
THE DISTRIBUTIVE PROPERTY
Addition is specific for like terms;
however, like and unlike terms can be multiplied using the
multiplication rules.
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Use the
distributive property to
multiply addition problems by a numerical term or
variable term.
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xi. SOLVING EQUATIONS Solving equations is an
algebraic process that isolates the variable in an equation for the purpose
of finding the value for the variable that balances or satisfies the
equation.
The solution to the equation
is the number that balances the equation when substituted for the variable.
In the equation x + 2 = 8, isolating the variable
results in the transformed equation x = 6, which is the solution. The solution to an equation is often written in
set notation as { 6 } or { x | 6 }.
xiii FOUR STEPS TO SOLVE AN
EQUATION
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The first step to solve an
equation is to simplify the expressions on both sides of the equals sign.
Follow the Order of
Operations Agreement when simplifying the expressions. Always
simplify using
multiplication before
simplifying using
addition by combining like terms.
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The second step to solve an
equation is to isolate the variable term, using the addition property of
equations.
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The third step to solve an equation is to
isolate the variable, using the multiplication
property of equations.
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The fourth step to solve an equation is to check the
solution of the equation by substituting the value calculated in the given equation for the variable in
the original equation. A correct solution will satisfy both expressions on
each side of the equals sign; therefore balancing the equation.
xiv.
THREE EXAMPLES TO SOLVE AN EQUATION
EXAMPLE 1
EXAMPLE 2
EXAMPLE 3
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EXAMPLE 1 |
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EXAMPLE 2
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EXAMPLE 3
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xvi. TRANSLATING ENGLISH INTO MATH
Problem solving often requires translating phrases into
mathematical symbols.
xvii. Translating Tips
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Read and translate from left to right except for "less than"
phrases.
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* indicates the word or phrase that must be translated from
right to left
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Use common sense.
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The word "and" does not mean addition. The word "and" is
used to separate the objects in the mathematical operation.
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"Is" indicates the equals sign
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"of" indicates multiplication
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word or phrase |
written in English |
written in math |
| sum of |
the sum of 2 and 3 is 5 |
2 + 3 = 5 |
| total of |
the total of 3 and 4 is 7 |
3 + 4 = 7 |
| more than |
12 more than 4 is 16 |
12 + 4 = 16 |
| increased by |
–4 increased by 13 is 9 |
– 4 + 13 = 9 |
| difference between |
the difference between 5 and 1 is 4 |
5 – 1 = 4 |
| subtracted from * |
7 subtracted from 4 is –3 |
4 – 7 = – 3 |
| less than * |
6 less than 3 is –3 |
3 – 6 = – 3 |
| less |
6 less 3 is 3 |
6 – 3 = 3 |
| product of |
the product of 5 and 4 is 20 |
5 • 4 = 20 |
| twice (multiply by 2) |
twice 6 is 12 |
2(6) = 12 |
| of (used with fractions) |
⅓ of 30 is 10 |
⅓ (30) = 10 |
| percent of |
12% of 16 is 1.92 |
.12 (16) = 1.92 |
| quotient of |
the quotient of –42 and 6 is –7 |
– 42 ÷ 6 = – 7 |
| divided by |
56 divided by –7 is 8 |
56 ÷ – 7 = 8 |
| ratio of |
the ratio of 3 to 4 |
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xix. More Translating Examples
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xx. WORD PROBLEMS
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Consider the following examples:
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If whole milk costs 25 cents more per gallon than skimmed milk,
express the cost of a gallon of whole milk in terms of the cost of a gallon of
skimmed milk.
Solution: Since the cost of skimmed
milk could be any amount greater than zero, use a variable to represent the
cost and let x = the cost of skimmed milk. Since 1 gallon of whole milk
costs 25 cents more than 1 gallon of skimmed milk, let x + 0.25 =
the cost of whole milk. Twenty five cents expressed in dollars
is .25.
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Jerry has 6 less than one-third as many baseball cards as Bill.
Write an algebraic expression for the number of Bill's baseball cards and a
expression for the number of Jerry's baseball cards.
Solution: Since Bill could have any
number of baseball cards, use a variable to represent the number of his baseball
cards. Let x = the number of Bill's cards. Therefore, ⅓x =
one-third the number of Bill's baseball cards. Since Jerry has 6 less
baseball cards than ⅓ of Bill's baseball cards, then ⅓x – 6 = the number of
Jerry's baseball cards.
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A chemistry teacher wants to build
a shelf that will hold 10 reagent bottles. If each bottle has a diameter
of d, how long should the shelf be?
Solution:
Let d = the length of each bottle. Since a shelf will hold 10 bottles then
10d = length of each shelf.
xix. Four Steps to Solving Word
Problems
| Consider the following example:
The main body of a missile is 9
times as long as the nose cone. The entire missile is 100 feet in length.
How long is the nose cone?
Solution in 4 Steps: |
| STEP ONE--Label the Variable |
Choose a variable to represent the number you want to find and use the
variable in describing all other numbers. |
Since the nose can be any number
greater than 0, let x =
Length of the nose cone. Since the
problem tells us that the main body is 9 times as long as the nose cone,
let 9x
= Length of the main body |
| STEP TWO--Write the Equation
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Use the facts given in the problem to
write the equation.
Show that the sum of the two lengths labeled in
step one are equal to 100. |
Length
of nose cone +
Length
of main body = 100
x + 9x
= 100 |
| STEP THREE--Solve the Equation |
Simplify each side of the equals
sign,
isolate the variable term, and
isolate the variable |
x + 9x = 100
10x = 100
x
= 10 |
| STEP FOUR--CHECK
YOUR ANSWER Determine if the answer
is sensible. |
Use the words in the problem to
determine if the answer is sensible.
Length of the nose cone; x = 10
Length of the main body;
9x
= 9(10) |
x
+ 9x = 100
10 +
9(10)
= 100 |
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i. COMMUTATIVE PROPERTY FOR ADDITION AND MULTIPLICATION
The commutative property allows numbers to be added or
multiplied in any order.
For example: 2 + 3 = 3 + 2 and 2 ( 3 ) = 3 ( 2 ).
The commutative property of addition may be stated:
For every number a and b; a + b = b + a.
The commutative property of multiplication
may be stated:
For every number a and b; ab = ba.
The commutative property does not apply to division problems nor
to subtraction problems.
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ii. ASSOCIATIVE PROPERTY FOR ADDITION AND MULTIPLICATION
The Associative Property allows regrouping adjacent numbers
in addition and multiplication problems.
For example, (1 + 2) + 3 = 1 + (2 + 3) and
2 × (3 × 4) = (2 × 3) × 4.
The associative property of addition may be stated:
For every number a, b and c; a + (b + c) = (a + b) + c.
The associative property of multiplication
may be stated:
For every number a, b and c; a(bc) = (ab)c
The associative property does not apply to division problems nor
to subtraction problems.
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iii. IDENTITY PROPERTIES FOR ADDITION AND MULTIPLICATION
0 + 4 = 4 illustrates the identity property of addition.
Zero was added to four and the four did not change. The additive property of 0 or the additive identity
may be stated:
For every number a; 0 + a = a.
1 × 4 = 4 illustrates the identity property of multiplication.
Four was multiplied by one and the four did not change. The multiplicative property of 1 or the
multiplicative identity may be stated:
For every number a; 1• a =
a.
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iv.
INVERSE PROPERTIES
FOR ADDITION AND MULTIPLICATION
The additive inverse may be
stated:
For every number a; a + (–a)
= 0 and –a + a = 0.
A number added to its
opposite results in the
answer of 0.
The multiplicative inverse may be
stated:
For every number a; except a = 0, a • 1/a =
0 and 1/a • a = 1.
A number multiplied by its reciprocal
always results in the answer of 1.
To find the
reciprocal of a number exchange the number in the numerator with the number
in the denominator.
1/2 is the reciprocal of 2 because 1/2 ( 2
) = 1
2/3 is the reciprocal of 3/2 because 2/3 (
3/2 ) = 1
-4 is the reciprocal of -1/4 because -4 (
-1/4) = 1
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v.
THE MULTIPLICATION PROPERTY OF ZERO
Zero times any number is zero. The multiplicative property of
0 may be stated: For every number a;
0 • a = 0.
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vi. DIVISION BY ZERO
The
multiplication property of zero
affects division by zero.
6 ÷ 3 = 2 because 2 • 3 = 6;
6 ÷ 0 does not exist because 0 • any number
= 0, not 6.
This also means that a fraction having a
denominator of 0 contradicts the multiplication property of 0, since the
fraction bar is also a symbol for division.
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vii. DISTRIBUTIVE PROPERTY
The distributive property allows an
addition problem to be part of a multiplication problem. Multiplication is
distributed over the addition.
The distributive property may be stated:
For every number a, b, and c; a(b
+ c) = ab + bc.
Use the distributive property to
multiply addition problems that cannot be simplified because the
addends are not like terms. For example: 2( 3x
+ 6 ) = 2(3x) + 2(6) = 6x + 12
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viii. REFLEXIVE PROPERTY
The reflexive property states that
any number is equal to itself. This is the simplest of all the properties.
The reflexive
property may be stated:
For every number a; a =
a.
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ix. SYMMETRIC PROPERTY OF
EQUALITY
The symmetric property of equality
allows equality to be reversed.
The symmetric
property of equality may be stated; For every number a and
b; if a = b, then b = a.
3x + 5 = 8 and 8 = 3x + 5 are the same because of the symmetric
property of equality.
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x. TRANSITIVE PROPERTY OF EQUALITY
The transitive property of equality allows for the transfer of
equality. Two amounts are equal if they both are equal to the same third
amount.
The transitive property of equality may be
stated:
For every number a, b, and c; if a = c and b = c, then a = b.
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xi. ADDITION PROPERTY OF EQUATIONS
The Addition Property of Equations allows the same amount to be
added to both sides of the equals sign in an equation. The solution to the
equation does not change. Only the appearance of the equation changes.
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xii. MULTIPLICATION PROPERTY OF EQUATIONS
The Multiplication Property of Equations allows both sides of
the equals sign in an equation to be multiplied by the same amount. The
solution to the equation does not change. Only the appearance of the
equation changes.
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