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Key Words/Topic
and Assignments
1.1
Numerical
Expressions
Topic 1 Guided Notes
Variables and Expressions
Information, Definitions, Solutions
New Terms
Numerical A mathematical ________ that consists of numbers and operations
Expression __________. (ex. 4 + 15)
Evaluate To ________ or _________.
Equivalent Expressions that always have the same _______.
Expressions
Sum The answer to an _____________ problem.
Difference The answer to an _____________ problem.
Product The answer to an _____________ problem.
Quotient The answer to an _____________ problem.
Review Terms
Order of This is the order in which operations should be done in an expression.
Operations Operations inside parentheses are done first, followed by exponents.
Then, multiplication and division are done in order moving from left to
right. Addition and subtraction are done last, also moving from left to
right.
Today’s Concept You’ve been using numerical expressions for a long time. As long as
your expression has numbers and at least one operation (+, -, •, ÷)* then
it’s a numerical expression. An expression does NOT have an = sign. If
you add an equal sign to an expression, the expression becomes an
equation.
*Notice I used a • for multiplication instead of an X. As you move into
algebra, you want to avoid using an X as a multiplication symbol, so as
not to confuse the symbol with the variable X.
You should be able to take a basic word phrase and turn them into
1.1
Numerical numerical expressions. In order to do this, and all of math, well you
Expressions need to understand and use the proper vocabulary.
continued
There are many word phrases you can turn into numerical expressions.
Group Work
Here are a few:
HOMEWORK:
P. in textbook.
Word Phrase
12 increased by 2
Numerical Expression
12 + 2
The difference of 24 and 13
24 – 13
The product of 12 and 6
12 • 6
The quotient of 45 and 9
45 ÷ 9
Some words in the expression provide clues that tell you what operation
to use.
Addition clues: Sum, increased by, more, added
Subtraction clues: Difference, decreased by, less
Multiplication clues: Product, times
Division clues: Quotient, divided
Numerical expressions are called equivalent expressions, if after you
evaluate the expressions the values are equal.
For example: 4(120 ÷ 6) is equivalent to (2 • 12) + 56
Numerical expressions can vary in length. In order to know how to
evaluate (find or solve), or simplify, the expression you need to
remember the order of operations.
ORDER OF OPERATIONS REVIEW
Many of you already memorized PEMDAS. PEMDAS is a good tool,
but only if you remember that it is not perfect as written!
How do I solve (52 + 3 *2) – 14 ÷ 2?
Here’s how PEMDAS works.
1. Complete operations within parentheses(P) first. If there are
multiple operations inside parentheses, you must follow PEMDAS
inside the parentheses.
Since 52 + 3 *2 is inside parentheses, solve this first.
1.1
Numerical 2. Next do any exponents (E).
Expressions
continued Solve 52 first and then REWRITE the problem with what you know.
52=25 so (25 + 3 *2) – 14 ÷ 2 is the revised problem.
3. Now you multiply(M) and divide(D) moving left to right. You
DON’T only do multiplication first just because the M is first.
We are still inside the parentheses so solve 3*2 and then REWRITE
the problem with what you know. 3*2=6 so (25 + 6) – 14 ÷ 2 is the
revised problem.
4. Next you add(A) and subtract(S) moving left to right. You
DON’T only do addition first just because the A is first.
We are still inside the parentheses so solve 25+6 and then
REWRITE the problem with what you know. 25+6 =31 so 31 – 14 ÷
2 is the revised problem.
Now that we are done with the operations within the parentheses, we
can finish the problem by following PEMDAS for what is left.
14÷2=7
31-7=24!
Key Words/Topic
and Assignments
1.2
Algebraic
Expressions
Information, Definitions, Solutions
New Terms
Algebraic A ________________ phrase that consists of ___________, numbers,
Expression and ____________ symbols. (3x + 4)
Variable A letter that represents an ____________ value. (ex. x, y, a)
Term A number, ____________, or the product of a number and one or
________ variables. (ex. z, 3y, and 12)
Constant A _______ that contains (is made of) a number. (ex.12)
Coefficient Is the ____________ part of a term that contains the variable. (ex.in 3x,
3 is the coefficient)
Review Terms
Quantity Something you can measure.
Today’s Concept Variables are usually letters that represent (stand in for) a number.
Every algebraic expression must have at least one variable and at
least one operation. Algebraic expressions can also, and very often do,
have numbers. Sometimes algebraic expressions are called variable
expressions.
You can turn any numerical expression into an algebraic expression by
using at least one variable.
To determine what the variable expression should be
1. Read the situation
2. Determine what operation the situation is referring to
3. Select a letter, or letters, to represent the variable or variables.
4. Combine the variable, the operation, and any other terms in the
correct order.
Some words provide clues that tell you what operation to use.
Addition clues: Sum, increased by, more, added
Subtraction clues: Difference, decreased by, less
Multiplication clues: Product, times
Division clues: Quotient, divided
Remember that a quantity is something you can measure. A variable
1.2
Algebraic quantity is a measurement that can vary (change).
Expressions
continued Each comic book I purchase costs $3.99 (a quantity), but the number of
comic books I purchase each week changes (a variable quantity).
Group Work
Key Words/Topic
and Assignments
1.3
Writing
Algebraic
Expressions
Information, Definitions, Solutions
New Terms
Factors Are __________ to give a product.
Power A number _____________ using an exponent.
Exponent A number that _______ how many times a base is used as a
____________.
Bar Diagram A way to represent part to whole ___________________.
Review Terms
Algebraic A mathematical phrase that consists of variables, numbers, and
Expression operation symbols. (3x + 4)
Variable A letter that represents an unknown value. (ex. x, y, a)
Term A number, variable or the product of a number and one or more
variables. (ex. z, 3y, and 12)
Constant A term that contains (is made of) a number. (ex.12)
Coefficient Is the number part of a term that contains the variable. (ex.in 3x, 3 is the
coefficient)
Today’s Concept
You are often presented with situations where you don’t know
everything. If the information you don’t yet know is mathematically
based, you maybe able to construct an algebraic expression to find the
unknown information (quantity).
Just like when you constructed numerical expressions from word
phrases, you need to be able to construct algebraic expressions from
word phrases. The technique is the same. You need to look for clues
that will help you construct the expression.
Word Phrase
12 increased by a number
Group Work
HOMEWORK:
Algebraic Expression
12 + x
The difference of a number and 13
y – 13
The product of a number and 6
n•6
The quotient of 45 and a number
45 ÷ x
Key Words/Topic
and Assignments
1.4
Evaluating
Algebraic
Expressions
Information, Definitions, Solutions
New Terms
Review Terms
Substitution Substitution replaces the variable with a number
Algebraic A mathematical phrase that consists of variables, numbers, and
Expression operation symbols. (3x + 4)
Variable A letter that represents an unknown value. (ex. x, y, a)
Term A number, variable or the product of a number and one or more
variables. (ex. z, 3y, and 12)
Constant A term that contains (is made of) a number. (ex.12)
Coefficient Is the number part of a term that contains the variable. (ex.in 3x, 3 is the
coefficient)
Today’s Concept Once you have an algebraic expression, you can replace the variable
with a value (number) and then evaluate the expression.
Take the expression 3(x + 2).
1. Pick a value for x. Let’s pick 5
2. Substitute the x for 5; 3(5+2)
3. Use the order of operations to simplify
4. 3(7) = 21
Group Work
HOMEWORK:
Key Words/Topic
and Assignments
1.5 Expressions
with Exponents
Information, Definitions, Solutions
New Terms
Power A number _________ using an _____________.
Base The repeated ___________ of a number written in exponential form. (ex
in 3⁴ the 3 is the base)
Exponent A number that shows how many times a _________ is used as a
____________. (ex in 3⁴ the 4 is the base)
Review Terms
Today’s Concept
Exponents help to write repeated multiplication efficiently. Instead of
7•7•7•7•7, you can write 75. They have equal value. A power has two
components, the base and the exponent.
The base is the number that is multiplied by itself (a repeated factor)
however many times the exponent tells us. So if we have 75. The 7 is the
base and the 5 is the exponent or power.
ERROR ALERT!!! NEVER multiply the base times the exponent.
75 ≠ 35, 75 = 16,807!
When solving numerical, or algebraic, expressions with exponents, you
still need to follow the order of operations. Remember that exponents
come after working through all of the grouping symbols like
parentheses.
Take 43 + (62 – 6)
1. First complete everything within the parentheses. Remember to
follow the order of operations within the grouping symbol. As
you solve each step REWRITE problem with your new
information. 43 + (6•6-6)
2. 43 + (36 – 6)
3. 43 + 30
4. 4•4•4 = 64
5. 64 + 30 = 76
Take 2x2 ÷ 3, if x = 6
1. First substitute 6 for x. 2(62) ÷ 3
2. 2(36) ÷ 3
3. 72 ÷ 3 = 24
Group Work
Homework