Transcript File

Learning Target:
I can…
Write algebraic expressions from word problems
Introduction to Algebra
Variable:
A ______________________ that
stands for a ________________
ALGEBRAIC EXPRESSION
Uses numbers, variables, and operational
symbols
Key Words for Operational Symbols:
Each __________
More __________
Less ___________
Share Equally ____________
Example 1
At a ballpark, team hats are sold for $15 each.
Let c = the number of hats. Write an algebraic
expression for the cost of a number of team
hats.
2) Jake has four more books than Sarah. Let b =
the number of books Sarah has. Write an
algebraic expression for the number of books
Jake has.
You Try!
3. Every person in Mrs. Graf’s class brings two
pencils to take a test. If there are c students
in the class, write an algebraic expression to
find the number of pencils total.
4. Lena is exactly 3 years older than her brother,
Peter. If p represents Peter’s age, write an
algebraic expression to represent Lena’s age.
5. Elaina and her friends went out to dinner and
decided to split the bill evenly. The cost for
dinner was $56.75. If there were n number of
people out to dinner, how much did each
person pay?
OAA Example
6)
Two Step Expressions
1) To get into the county fair, Patricia must pay a
$5 entrance fee and $2 per ride. Write an
expression to represent the total Patricia
spends at the fair.
2) A group of 10 people at the Village decides
to split their bill evenly. If b represents the bill
and each person says they will throw in $1 for
a tip, write an expression to represents the
amount each person spends.
3)
Jill charges a base rate of $25 when she mows
a lawn. She also charges $6 per hour she
mows. If h represents the number of hours
she mows, write an expression to represent
the amount she makes mowing her neighbors
lawn.
4)
Jasmine wants to rent a bike while she is on
vacation. The rental fee is $8.00 plus an
additional $2.50 for every hour the bike is
rented. If h represents the number of hours
the bike is rented, write an expression to
represent the total amount spent.
5) The length of a rectangle is 6 cm.
Write an expression to represent the
perimeter of the rectangle, using w to
represent the width.
OAA EXAMPLE
Tax, Tip, Discount Expressions
1) Use p to represent the original cost of an
item. Write the expression for the total
amount spent if the price is discounted 20%.
You Try!
2) Use c to represent the original cost of an
item. Write the expression for the total
amount spent if the price is discounted 25%.
3) Use c to represent the cost of an item. Write
an expression to represent total spent if the
item is bought with 7% tax.
4) Use b to represent the bill at Bob Evan’s.
Write an expression to represent total spent if
you also pay a 15% tip to the waiter.
EXIT
Write the expression for the following situations:
1) Total candy if there are k kids in a class and each
kid brings 3 pieces.
2) Total amount of money made by a waitress who
makes $3 per hour plus $25 in tips. (h
represents hours)
Learning Target:
I can…
Write and evaluate expressions
To Evaluate an Algebraic Expression
1. Rewrite equation and plug in a value for
each variable
2. Simplify
Example 1
Evaluate 3x² + 4 + y when x = -2 and y = 5
2) Evaluate 4a² + 3a + 1 when a = -2
3) Evaluate 2a² + 3b + c when
a = -3, b = 4 and c = 5
4)
Write and Evaluate
Remember key words for writing expressions:
Each __________
More __________
Less ___________
Split Evenly ____________
Write and Evaluate
1) To get into the county fair, Patricia must pay a
$5 entrance fee and $2 for each ride. Write
an expression to represent the total Patricia
spends at the fair.
Write the expression _________________
If Patricia went on 10 rides, how much did
she spend?
Evaluate ___________________________
Write and Evaluate
2) Elaina and her friends went out to dinner and
decided to split the bill evenly. The cost for
dinner was $56.75. If there were n number
of people out to dinner, how much did each
person pay?
Write the expression _________________
If 8 people were out to dinner, how much did
each person pay?
Evaluate ___________________________
Exit
If x = -5, what is x² + 7?
Name_________________________
RED
Evaluate the following expressions for x = 4 Evaluate the following expressions for x = -3
1)
x – 10
1)
x – 10
Rewrite expression _____________
Rewrite expression _____________
Answer ________________
Answer ________________
2)
2)
3x
3x
Rewrite expression _____________
Rewrite expression _____________
Answer ________________
Answer ________________
3)
3)
-x
-x
Rewrite expression _____________
Rewrite expression _____________
Answer ________________
Answer ________________
4)
4)
5x + 4x
5x + 4x
Rewrite expression _____________
Rewrite expression _____________
Answer ________________
Answer ________________
5)
5)
32/x
30/x
Rewrite expression _____________
Rewrite expression _____________
Answer ________________
Answer ________________
6)
6)
3/x + .25
3/x + .25
Rewrite expression _____________
Rewrite expression _____________
Answer ________________
Answer ________________
7)
x² – 9
7)
x² – 9
Rewrite expression _____________
Rewrite expression _____________
Answer ________________
Answer ________________
Evaluate the following expressions for
x = -2 and y = 5
1)
x–y
Rewrite expression _____________
Answer ________________
2)
3x + 2y
Rewrite expression _____________
Answer ________________
3)
-x – 3y
Rewrite expression _____________
Answer ________________
4)
5x + 4y + 10
Rewrite expression _____________
Answer ________________
5)
32/x + 15/y
Rewrite expression _____________
Answer ________________
6)
3/x + y/4
Rewrite expression _____________
Answer ________________
7)
x² + y
Rewrite expression _____________
Answer ________________
4)
5) Sal is having a party and having people
sit at a long line of tables. Each table can
sit 4 people, plus one person cat sit at
each end of the line of tables, as shown.
If there are 12 tables, how many people can sit down?
6)
If there are 20 tables, how many people can sit down?
7)
Name_________________________
YELLOW
3) Sal is having a party and having people sit at a long line of
tables. Each table can sit 4 people, plus one person cat sit at
each end of the line of tables, as shown.
1) Evaluate the following expressions for x = 4
A. x – 10
B.
3x
C.
-x
D.
5x + 4x
E.
32/x
F.
3/x + .25
G.
x² – 9
If there are 12 tables, how many people can sit down?
2)
If there are 20 tables, how many people can sit down?
Convert the following temperatures
(round to the nearest degree if
necessary)
80°F = ______°C
4)
The cost to ship a package is $2.79 plus an additional $0.38 per
pound. Write an expression to find the total spent.
50°F = ______°C
75°F = ______°C
60°F = ______°C
100°F = ______°C
How much would it cost to ship an 8 pound package?
5)
6)
8)
7)
Name_________________________
GREEN
1)
Meg and Tom were both running for Student Council President.
Meg received 50 more than twice the amount that Tom received.
If Tom received v votes, write the expression for the amount Meg
received.
Evaluate the expression if Tom received 30 votes.
3)
Sara is jogging around her neighborhood. She is running at a rate
of .1 miles per minute. Write an expression to represent her
total distance after m minutes.
How long will it take her to run 3 miles?
2)
Jason has 50 gallons of water in his bathtub. The bathtub drains
at 4 gallons per minute. Write the expression Jason can use to
find the total amount of water left in the bathtub after m
minutes.
Evaluate the expression if Jason has let the water drain for 180
seconds.
4) Sal is having a party and having people sit at a long line of tables.
Each table can sit 4 people, plus one person cat sit at each end of the
line of tables, as shown.
Write an expression to determine how
many people can sit at x number of tables.
If there are 12 tables, how many people can sit down?
If there are 20 tables, how many people can sit down?
If 70 people attend the party, how many tables will he need?
5)
6)
8)
7)
Evaluate the following expressions for
for x = -4 and y = -2
A.
x–y
B.
3x + 2y
C.
-x⁴ + y²
D.
5x + 4y
E.
32/x + y
F.
3/x + .25y
G.
x² – 3y
If x = 4, solve problems 1-10
1. x – 10
2. 3x
3. -x
4. 5x + 4x
5. 32/x
6. 3/x + .25
7. X²
8. x² – 9
9. √x
10.4x³
Do the rest on your own and get
checked!
Learning Target:
I can…
Solve one and two step equations
Inverse Operations
The inverse of each operation cancels it out.
Inverses you will use:
Add
~
Subtract
Multiply ~
Divide
Squared ~
Square Root
Multiply by a fraction (ex: ¾)
~Multiply by its reciprocal (4/3)
Solving Equations
1. Separate the equation
2. Start on same side of variable, but
furthest from the variable
3. Inverse operations
4. Isolate the variable
Examples:
One step: 2x = 46
Two step: 26 + 2x = 46
31.5==57.9
9c
a –x 61.3
+ 36 = 106
97.5
= f/15
x² = 44.89
You try
1)
2x = -7
2)
6.5 = ½x
3)
x/4 = 16
4)
6 = -¾x
5)
x  6.7
OAA Example
4.5a = 17.1
a=?
Solve for a.
10 +
a
4
= 15
4r + 6 = 14
-11 = -5b – 6
y
-7
+ 8 = 19
EXIT
Solve 4p + 27 = 43
Learning Target:
I can…
Show how to solve equations
1.75 +
1.75 +
1.75 – 1.75 +
d
4
=3.2
d
= 3.2.
4
d
4 = 3.2 – 1.75
d
= 1.45
4
(4) d = (4)1.45
4
d = 5.8
The cost of a bag of doughnuts is $5.80.
Subtract 1.75 from each side.
Simplify.
Multiply each side by 4.
Simplify.
Three Step
2x² - 13 = 19
-60 = -3a² + 15
93 = 4x² – 7
EXIT
Solve for a.
3a² – 35 = 397
-x = 17
10 = 1 - x
12 – x = -3
25 – 3x = 2.5
16 – 2x = 9
5 – 3x = 11
4 – 2y = 7
Learning Target:
I can…write equations to solve word problems
Word Problems - Example
Johnny makes $12.75 per hour at his job. If he
made $459.00 this week, how many hours did
he work?
Identify x ______________
Equation _______________
Answer ________________
Word Problems – Example 2
Justin bought 2 apples and 1 pear for $4.00. The
pear was $1.78. Write an equation and solve
to find the cost of each apple.
Identify x ______________
Equation _______________
Answer ________________
1.
Mark bought 4 pens and 1 notebook for $3.25.
The notebook was $2.25. Write an equation
and solve to find the cost of each pen.
Identify x ______________
Equation _______________
Answer ________________
2.
Sara bought 2 bags of chips and a coke for
$7.50. The coke was $1.50. How much was
each bag of chips?
Identify x ______________
Equation _______________
Answer ________________
3.
The fair costs $3 to get in and $2 for every ride.
If John spent $21, how many rides did he go
on?
Identify x ______________
Equation _______________
Answer ________________
4.
Sara is jogging around her neighborhood. She is
running at a rate of .1 miles per minute. How
long will it take her to run 3 miles?
Identify x ______________
Equation _______________
Answer ________________
5.
Logo Shirts, Inc. is going to charge Urbana Junior
High $5 for each t-shirt they make for the
school. The company will also charge the
school a flat fee of $35 to set up the design. If
they spent $500 total how many shirts did they
buy?
Identify x ______________
Equation _______________
Answer ________________
6.
Mike, who has 6 video games, has half as many
games as Paul. How many does Paul have?
Identify x ______________
Equation _______________
Answer ________________
7.
Meg received 90 votes for Student Council
President, which was 50 more than twice the
amount that Tom received. How many did
Tom receive?
Identify x ______________
Equation _______________
Answer ________________
8.
The 347 students who listed soccer as their
favorite sport were 13 less than three times
the number of students who listed basketball
as their favorite sport. How many listed
basketball as their favorite sport?
Identify x ______________
Equation _______________
Answer ________________
9.
331 students went on a field trip. Six buses were
filled and 7 students traveled in cars. How
many students were in each bus?
Identify x ______________
Equation _______________
Answer ________________
You bought a magazine for $5 and four erasers.
10. You spent a total of $25. How much did each
eraser cost?
Identify x ______________
Equation _______________
Answer ________________
You bought a magazine for $5 and four erasers.
You spent a total of $25. How much did each
eraser cost?
A. 5x + 25 = 20
B. 5x + 4 = 25
C. 4x + 5 = 25
D. -4x + 5 = 25
EXIT
Jacqueline bought a CD for $16.95 and eight
DVDs. The total cost was $52.55. Write the
equation and find the cost of each DVD.
Equation: ___________________
Answer:_________________
Logo Shirts, Inc. is going to charge General
Sherman $6 for each t-shirt they make for the
school. The company will also charge the
school a flat fee of $35 to set up the design.
The school spent $575 all together. Which
equation represents this situation?
A. 35x + 575 = 6
B. 6x + 575 = 35
C. -6x + 35 = 575
D. 6x + 35 = 575
331 students went on a field trip. Six buses were
filled and 7 students traveled in cars. How
many students were in each bus?
a. 6x + 7 = 331
b. 7x + 6 = 331
c. 331x + 6 = 7
d. 7 + 331x = 6
Jasmine wants to rent a bike while she is on
vacation. The rental fee is $8.00 plus an
additional $2.50 for every hour the bike is
rented. If she spent $20.50, how many hours
did she rent the bike?
Writing Functions
1. Look at the first pair of x,y
values
2. Consider different ways to
get from x to y
3. See what works for the rest
of the table
Function: y = 2x
x
y
3
6
4
8
5
10
What is the funtion?
a. y = 3x
b. y = x + 3
c. y = x + 4
d. y = 2x
x
y
-5
-2
-1
2
1
4
3
6
Describe the rule of
the table.
a. y = 3x
b. y = x + 3
c. y = x + 4
d. y = 2x
x
y
2
6
3
9
4
12
5
15
Describe the rule of
the table.
a. y = x - 4
b. y = 4x
c. y = x + 4
d. y = -4x
x
y
2
-8
3
-12
4
-16
5
-20
When a table needs more than one operation
Tricks:
1. It typically will be
multiply then add or
multiply then subtract
2. …so look for what you
can multiply to get close
to y
3. Example:
Rule: y = 4x + 1
x
y
1
5
2
9
3
13
4
17
When the rule needs more than one
operation
x
y
1
2
3
8
5
14
7
20
9
?
10
?
Rule: y = 3x – 1
What is the rule?
a. y = 2x
b.y = 2x + 2
c. y = 2x – 2
d.y = x + 2
x
y
1
4
2
6
3
8
4
10
What is the rule?
a. y = 3x
b.y = 2x + 2
c. y = 3x – 2
d.y = 3x + 1
x
y
2
7
4
13
6
19
8
25
x
1
2
3
4
a.
b.
c.
d.
y = 2x
y=x+2
y = 2x + 2
y = x²
y
1
4
9
16
Writing rules for patterns
Example:
Write a rule for the nth term
in the sequence 2, 8, 14, 20,
…
1. Make a table that pairs the
term number and value
2. Find the rule:
3n-1
n
value
1
2
2
8
3
14
4
n
20
Problem of the Day (Thursday)
Creating Function Tables
x
Rule
(2x + 1)
y
(x,y)
Graph the function y = 5x - 3
x
5x - 3
y
(x,y)
1
5(1) - 3
2
(1,2)
2
3
4
Graph the function y = ½x + 2
x
½x + 2
y
(x,y)
1
½(1) + 2
2.5
(1,2.5)
2
3
4
y = 6x
Graphing Functions
Linear Function: Makes a straight line
Nonlinear Fuction: Does not make a straight line
Is the function linear or nonlinear?
y = 2x + 1
x
2x + 1
y
(x,y)
1
2(1) + 1
3
(1,3)
2
3
4
Linear
Is the function linear or nonlinear?
y = x²
x
x²
y
(x,y)
1
1²
1
(1,1)
2
3
4
Nonlinear
CPS Practice Problems!
Linear/Nonlinear Function activity
Learning Target:
I can…
Find the nth term of a sequence
What are the first 5 terms of the
sequence 2n?
A.
B.
C.
D.
2, 4, 6, 8, 10
1, 2, 3, 4, 5
2, 4, 8, 16, 32
1, 3, 5, 7, 9
What are the first 4 terms of the
sequence n²?
A.
B.
C.
D.
2, 4, 6, 8
1, 2, 3, 4
1, 4, 9, 16
1, 3, 5, 7
What are the first 4 terms of the
sequence n(n + 1)?
A.
B.
C.
D.
2, 4, 6, 8
2, 6, 12, 20
1, 3, 5, 7
2, 3, 4, 5
What are the first 4 terms of the
sequence n(n + 4) – 6?
A.
B.
C.
D.
-1, 4, 6, 8
2, 6, 12, 20
-1, 6, 15, 26
-1, 3, 4, 5
Find the first 4 terms of the
sequence -3n + 10
A.
B.
C.
D.
7, 4, 1, -2
13, 16, 19, 22
7, 9, 11, 13
9, 11, 13, 15
Patterns and Tables
Write an expression for the rule
for 2, 4, 6, 8, 10, . . . .
Make a table that pairs the term numbers and
the values in the sequence.
What is the rule for the nth term in the
sequence 3, 5, 7, 9…
a. n + 2
b. 2n
c. 2n + 1
d. 3n - 1
What is the rule for the nth term in the
sequence 3, 5, 7, 9…
a. n + 2
b. 2n
c. 2n + 1
d. 3n - 1
What is the rule for the nth term in the
sequence ½, 1, 1½, 2…
a.
b.
c.
d.
n/2
2n
½+n
n–½
What is the rule for the nth term in the
sequence 3, 7, 11, 15…
a.
b.
c.
d.
y = 3n
y = 4n – 1
y=n+4
y = 2n + 1
Write a rule for the picture
Write the rule.
How many squares will be found in the 6th figure?
Write the rule.
How many people will be found in the 10th row?
Write the rule.
How many smileys will be earned on day 9?
Learning Series Practice!
Problem of the Day (Thursday)
What is the rule?
a. y = 4x – 1
b. y = 4x
c. y = 2x + 2
d. y = 5x - 1
x
1
2
3
4
y
4
9
14
19