Transcript Lesson 1-3 PowerPoint - peacock

Algebra 2
Algebraic Expressions
Lesson 1-3
Goals
Goal
• To evaluate algebraic
expressions.
• To simplify algebraic
expressions.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Essential Question
Big Idea: Variables, Properties
• How can you use the properties of real numbers to
simplify algebraic expressions?
– Some mathematical phrases and real-world quantities
can be represented using algebraic expressions.
– Variables can represent variable quantities in real world
situations and in patterns.
– The properties that apply to real numbers also apply to
variables that represent them.
Vocabulary
•
•
•
•
•
Evaluate
Term
Coefficient
Constant Term
Like Terms
Modeling Words with
Algebraic Expressions
• To represent real-world situations
containing variable quantities with
mathematical phrases you must translate
word phrases into algebraic expressions,
looking for words that describe
mathematical operations (addition,
subtraction, multiplication, division).
What words indicate a particular
operation?
Addition
• Sum
• Plus
• More than
• Increase(d) by
• Perimeter
• Deposit
• Gain
• Greater (than)
• Total
Subtraction
• Minus
• Take away
• Difference
• Reduce(d) by
• Decrease(d) by
• Withdrawal
• Less than
• Fewer (than)
• Loss of
Words for Operations - Examples
Words for Operations - Examples
What words indicate a particular
operation?
Multiply
• Times
• Product
• Multiplied by
• Of
• Twice (×2), double (×2),
triple (×3), etc.
• Half (×½), Third (×⅓),
Quarter (×¼)
• Percent (of)
Divide
• Quotient
• Divided by
• Half (÷2), Third (÷3),
Quarter (÷4)
• Into
• Per
• Percent (out of 100)
• Split into __ parts
Words for Operations - Examples
Words for Operations - Examples
Example:
Write an algebraic expression for each word phrase.
A. 9 less than a number w
9 less than a number w
w
–
9
w–9
B. 3 increased by the difference of p and 5
3 increased by the difference of p and 5
3
+
(p
–
5)
3 + (p – 5)
Your Turn:
Write an algebraic expression for each word phrase.
A. 88 times the difference of h and 4
88 times the difference of h and 4
88 •
(h – 4)
88(h – 4)
B. the quotient of a number f and 6
quotient of f and 6
f

6
f
6
Your turn:
1) m increased by 5.
2) 7 times the product
of x and t.
3) 11 less than 4 times a
number.
4) two more than 6
times a number.
5) the quotient of a
number and 12.
1) m + 5
2) 7xt
3) 4n - 11
4) 6n + 2
5)
x
12
Your Turn:
Match the verbal phrase and the expression
1.
Twice the sum of x and three
A. 2x – 3
D
2.
Two less than the product of 3 and x
B. 3(x – 2)
E
3.
Three times the difference of x and two
C. 3x + 2
B
4.
Three less than twice a number x
D. 2(x + 3)
A
5.
Two more than three times a number x
C
E. 3x – 2
Example:
Write a word phrase for the algebraic expression 9 – 3c.
9 – 3c
9 – 3
9 minus
•
c
the product of 3 and c
9 minus the product of 3 and c
Your Turn:
Write a word phrase for the algebraic expression 7 + 19b.
7 + 19b
7 + 19
•
b
7 plus the product of
19 and b
7 plus the product of 19 and b
Modeling a Situation
To model a situation with an algebraic
expression, do the following:
1. Identify the actions that suggest operations.
2. Define one or more variables to represent the
unknowns.
3. Represent the actions using the variables and
the operations.
Example:
You start with $20 and save $6 each week. What algebraic
expression models the total amount you save?
Identify action
Operations
starting
amount
Define
Variables
Let
Write
Expression
20
plus
w
amount
saved
times
number
of weeks
= the number of weeks.
+
6
•
w
The expression 20 + 6w models the situation.
Your Turn:
Write an algebraic expression to represent each situation.
A. the number of apples in a basket of 12 after
n more are added
Add n to 12.
12 + n
B. the number of days it will take to walk 100
miles if you walk M miles per day
Divide 100 by M.
Your Turn:
You had $150, but you are spending $2 each day.
What algebraic expression models this situation?
Answer: Let d = the number of days
150 – 2d
Definition
• Evaluate – To evaluate an expression is to
find its value.
• To evaluate an algebraic expression,
substitute numbers for the variables in the
expression and then simplify the expression.
Order of Operations
Rules for arithmetic and algebra
expressions that describe what
sequence to follow to evaluate an
expression involving more than
one operation.
Remember the phrase
“Please Excuse My Dear Aunt Sally”
or PEMDAS.
ORDER OF OPERATIONS
1.
2.
3.
4.
Parentheses - ( ) or [ ]
Exponents or Powers
Multiply and Divide (from left to right)
Add and Subtract (from left to right)
The Rules
Step 1: First perform operations that are within grouping
symbols such as parenthesis (), brackets [], and braces {},
and as indicated by fraction bars. Parenthesis within
parenthesis are called nested parenthesis (( )). If an
expression contains more than one set of grouping symbols,
evaluate the expression from the innermost set first.
Step 2: Evaluate Powers (exponents) or roots.
Step 3: Perform multiplication or division operations in order
from left to right.
Step 4: Perform addition or subtraction operations in order by
from left to right.
Example:
Evaluate the expression for the given values of the variables.
2x – xy + 4y for x = 5 and y = 2
2(5) – (5)(2) + 4(2)
Substitute 5 for x and 2 for y.
10 – 10 + 8
Multiply from left to right.
0+8
8
Add and subtract from left to right.
Your Turn:
Evaluate the expression for the given values of the variables.
q2 + 4qr – r2 for r = 3 and q = 7
(7)2 + 4(7)(3) – (3)2
Substitute 3 for r and 7 for q.
49 + 4(7)(3) – 9
Evaluate exponential expressions.
49 + 84 – 9
Multiply from left to right.
124
Add and subtract from left to right.
Your Turn:
Evaluate x2y – xy2 + 3y for x = 2 and y = 5.
(2)2(5) – (2)(5)2 + 3(5)
4(5) – 2(25) + 3(5)
20 – 50 + 15
–15
Substitute 2 for x and 5 for y.
Evaluate exponential expressions.
Multiply from left to right.
Add and subtract from left to right.
Definition
• Term – an expression that is a number, a
variable, or the product of a number and one
or more variables.
• Example:
– The expression 6x + yz – 7 has 3 terms, 6x, yz,
and 7. 6x and yz are variable terms; their values
vary as x, y and z vary. 7 is a constant term; 7 is
always 7.
Definition
• Coefficient – The numerical factor of a
term.
• Example:
– The coefficient of 3x2 is 3.
Definition
• Like Terms – terms in which the variables
and the exponents of the variables are
identical.
– The coefficients of like terms may be different.
• Example:
– 3x2 and 6x2 are like terms.
– ab and 3ab are like terms.
– 2x and 2x3 are not like terms.
Definition
• Constant Term– is a term with no variable.
– Constants terms are like terms.
– Example: in the expression - 4x + 3y2 – 5, the constant
term is – 5.
Example:
Recall that the terms of an algebraic expression are separated
by addition or subtraction symbols. Like terms have the same
variables raised to the same exponents. Constant terms are
like terms that always have the same value.
Simplifying Algebraic
Expressions
To simplify an algebraic expression, combine like terms by
adding or subtracting their coefficients. Algebraic expressions
are equivalent if they contain exactly the same terms when
simplified.
Remember!
Terms that are written without a coefficient have an
understood coefficient of 1.
x2 = 1x2
Simplifying Algebraic
Expressions
Example:
Simplify the expression.
3x2 + 2x – 3y + 4x2
3x2 + 2x – 3y + 4x2
Identify like terms.
7x2 + 2x – 3y
Combine like terms.
3x2 + 4x2 = 7x2
Example:
Simplify the expression.
j(6k2 + 7k) + 9jk2 – 7jk
6jk2 + 7jk + 9jk2 – 7jk
15jk2
Distribute, and identify like terms.
Combine like terms.
7jk – 7jk = 0
Your Turn:
Simplify the expression –3(2x – xy + 3y) – 11xy.
–6x + 3xy – 9y – 11xy
–6x – 8xy – 9y
Distribute, and identify like terms.
Combine like terms.
3xy – 11xy = –8xy
Example: Application
Apples cost $2 per pound, and grapes cost $3 per pound.
Write and simplify an expression for the total cost if you buy
10 lb of apples and grapes combined.
Let A be the number of pounds of apples.
Then 10 – A is the number of pounds of grapes.
2A + 3(10 – A)
= 2A + 30 – 3A
Distribute 3.
= 30 – A
Combine like terms.
Example: Continued
Apples cost $2 per pound, and grapes cost $3 per pound.
What is the total cost if 2 lb of the 10 lb are apples?
Evaluate 30 – A for A = 2.
30 – (2)
= 28
The total cost is $28 if 2 lb are apples.
Your Turn:
A travel agent is selling 100 discount packages. He makes $50 for
each Hawaii package and $80 for each Cancún package.
Write an expression to represent the total the agent will
make selling a combination of the two packages.
Let h be the number of Hawaii packages.
Then 100 – h is the remaining Cancun packages.
50h + 80(100 –h)
= 50h + 8000 –80h
= 8000 – 30h
Distribute 80.
Combine like terms.
Continued
How much will he make if he sells 28 Hawaii packages?
Evaluate 8000 –30h for h = 28.
8000–30(28) = 8000–840
= 7160
He will make $7160.
Assignment
• Section 1-3, Pg 22 – 24: #1 – 9 all, 10 – 54
even.