PowerPoint Presentation
Thinking AlgebraicallyAlgebra in Practical ApplicationHave you
everCounted your money to find out how much you spent?Calculated a
missing expense in your checkbook?Realized that you could still
solve a problem, even when there was missing information?8 - =
5Whats the missing number?This simple equation is an example of
algebra. In other words, if you started out with 8 dollars, and now
have 5 dollars, that means you spent 3 dollars. Algebra is a way to
solve problems when there is missing information.Some Key
VocabularyVariables: symbols used in place of unknown values.
Usually, a variable is a letter, such as x, y, a, or b. Constant: a
constant is a number whose value does not change. 5 + x = 8
Expressions and EquationsAlgebraic expressions are made of
constants, variables, and operations (+, -, x, / ).Expressions
never have an equal sign.
When two expressions are on opposite sides of the equal sign,
then you have an equation. 5 + x = 8Tackle the word problems like
this:Identify the variable: What are you trying to find?Identify
the known values: What do you want to know?Write a verbal model:
Use words and operations to define relationships.Write the
expression or equationJames collected $252.90 from his vending
machine route before lunchtime and then collected additional money
after lunch, making his total $426.10. Write an equation to find
out how much James collected after lunch.James collected $252.90
from his vending machine route before lunchtime and then collected
additional money after lunch, making his total $426.10. Write an
equation to find out how much James collected after lunch.Identify
the variable: What information is missing? The money James
collected after lunch
James collected $252.90 from his vending machine route before
lunchtime and then collected additional money after lunch, making
his total $426.10. Write an equation to find out how much James
collected after lunch.Identify the known values: Which value or
values are known? 252.90= the money James collected before
lunch$426.10=the total at the end of the dayJames collected $252.90
from his vending machine route before lunchtime and then collected
additional money after lunch, making his total $426.10. Write an
equation to find out how much James collected after lunch.Write a
verbal model:Money collected before lunch + money collected after
lunch = the total at the end of the dayJames collected $252.90 from
his vending machine route before lunchtime and then collected
additional money after lunch, making his total $426.10. Write an
equation to find out how much James collected after lunch.Write the
expression or equation. Using your verbal model, substitute the
known values and variables into the equation like this:252.90 + x =
426.10p. 104Last July, the Clark County animal shelter found homes
for 16 cats, two rabbits, a parrot, and an unknown number of
dogs.What is missing? Assign it a variable.We dont know how many
dogs found homes: dWhat are the known values? 16 cats, 2 rabbits,
and 1 parrotFor this problem, would you write an expression or an
equation? Write a verbal model# cats + # rabbits + # of parrots + #
of dogsWrite the expression:16 + 2 + 1 + dA scientist documents the
acceleration of a plane pulling out of a dive in different wind
speeds. In 17.5 miles per hour wind, the plane accelerates at 68.75
m/s squared. In 24.6 miles per hour wind, the plane accelerates at
an unknown rate. What are the known/unknown values?A scientist
documents the acceleration of a plane pulling out of a dive in
different wind speeds. In 17.5 miles per hour wind, the plane
accelerates at 68.75 m/s squared. In 24.6 miles per hour wind, the
plane accelerates at an unknown rate.
A scientist documents the acceleration of a plane pulling out of
a dive in different wind speeds. In 17.5 miles per hour wind, the
plane accelerates at 68.75 m/s squared. In 24.6 miles per hour
wind, the plane accelerates at an unknown rate. Write a verbal
model for the difference in acceleration: Acceleration in 17.5
m/hour wind acceleration in 24.6 m/hour windWrite an expression or
equation representing the difference in acceleration68.75 X
In a walkathon to fund cancer research through Liga
Puertorriquena Contra el Cancer (The Puerto Rican League Against
Cancer), comedian Ramon Rivera walked 80 miles and raised
approximately $85,000. Write a verbal model to find Ramon Riveras
funds raised per mile.Miles walked times the amount per mile =
total raisedWrite an expression or equation to find Ramon Riveras
funds raised per mile.80x = 85,000Note: whenever you see a number
next to a variable, like 80x, it means 80 times x. 24 3/16 n = 3
1/8Describe the equation in words:Twenty four and three sixteenths
minus n, equals three and one eighth.What is a real world problem
that you might use this equation to solve?All the soccer moms
bought 24 and 3/16 pizzas for the team. The soccer team ate a lot
of pizza. When they were done, there were only 3 and 1/8 pizzas
left. Using your word problem, write a different equation to solve
the problem. 24 3/16- 3 1/8 = n In her will, Mrs. Wilson divided
her money evenly among her five children. Write an expression or
equation to show how much money each child received. Identify the
unknown quantity and assign a variable. How much money Mrs. Wilson
left: mWould you use an expression or an equation to represent the
amount each child received? Why?An expression would be used because
we dont know what the divided assets would equal. Write the
expression or equation. Explain what operations you used and
why.m/5Divided evenly is a key word that signals divisionMarisa
bought a bowl for $24.99 plus 6.5% sales tax. Later, she went back
to the store to purchase salad tongs at the same sales tax
rate.Which operations will you need to write an expression
representing the total sales tax on both purchases? x and +How did
you determine what operations to use?Sales tax is always a
percentage of sales, and you multiply the rate by the purchase
price, so I knew x was to be used.The total sales tax, means that
we have to add the sales tax from the bowl with the sales tax from
the tongs, so + had to be used.Write an expression that represents
sales tax for both items. Use parentheses to indicate an operation
that needs to be done before other operations. .065(24.99 + x)
.065(24.99) + .065x A scientist gathers data on the time that
morning glories bloom locally throughout the year. The following
table represents a portion of her data:Write an expression to
represent the difference in median bloom time between May and June
in minutes. m - 336Write an expression for the difference in median
bloom time between March and April in minutes.358- 342 Compare it
to the first expression. How are they similar or different?Both of
them are simple subtraction problems, however, in the second
expression, all the values are known (constants). The difference in
median bloom time between April and May is the same as between May
and June. Write an expression to represent this:342 m = m -
336MonthMedian Bloom TimeMarch5:58amApril5:42MaymJune5:36Thinking
AlgebraicallyReasoning With Algebraic EquationsHave you
ever?Determined what you could afford at the supermarket?Calculated
the weekly deductions from your paycheck?Tried to solve a problem
using mental math but found it was too complex?An analogyAn
analogySome Key VocabularyCoefficient: The constant before a
variable that is multiplied with the variable.5x 259w-4yTerm: Is a
group of constants and variables multiplied together or just a
variable, or just a constant.5x100 x
Sherman earns $259 for every website he designs. How many
websites will he have to design in order to make $2,072?Understand
the Problem Sherman earns $259 for every website he designs. How
many websites will he have to design in order to make $2,072?What
are your known and unknown values? Assign a variable to the
unknown. Let w = number of websites
PlanSherman earns $259 for every website he designs. How many
websites will he have to design in order to make $2,072?Write an
equation:Use a verbal model: Amount per website x number of
websites = amount Sherman wants to makeInsert variables/constants
into your equation.259w = 2072AttackUse the inverse operation to
move the number away from the variable. (Also called isolating the
variable)What operations are present in this equation?259w =
2072What is the inverse operation of multiplication?
259w = 2072 259 259When you divide a term by a constant, you
divide the coefficient by the constant. 259w = 2072w = 8 259
259
CheckCheck by substituting the answer with the variable:259w =
2072259(8)= 20722072 = 2072Which one comes first?Write an
equation.Use substitution to check the result.Define the variable
and known values.Use inverse operations.Professor Gupta brought 32
student essays back to her classroom after correcting them, but she
only gave 28 back. The rest of the students were absent.Write and
solve a one-step algebraic equation to find how many students were
absent that day. Explain each step you take in solving this
problem. (Understand)Step 1: Identify your known/unknown variables
and assign a variable.What are you trying to find? How many
students were absent. Use variable s.What do you know? Graded
essays (32) and # handed back (28)(Plan)Step 2: Use a verbal model#
of essays handed back + # of students absent = total # of essays
gradedStep 3: Write an equation28 + s = 32(Attack)Step 4: Solve. Do
the inverse operation. 28 + s -28 = 32-28 s = 4(Check)Step 5:
Check28 + (4) = 32 32 = 32Thinking AlgebraicallyLinear Equations
with One VariableLinear EquationsCan be graphed as a straight
line
What is a graph and why does it matter? A Graph is a visual
representation of the relationship between numbers. All equations
can be graphed. Many real life relationships can be graphed. For
example, the amount of pumpkins sold at a farm is proportional to
time. The more days a farm is open, the more pumpkins they can
sell. So, time and number of pumpkins sold can be represented on a
graph like this:How can you recognize Linear Equations?Never have a
variable in the denominator of a fraction or divide by the
variable.Never have a variable with an exponent.
Which of the following equations is not a linear
equation?Choices:A. 3x - 5 = 8xB. 2 + 3x = 4C. (x + 2)2 = 6D. x =
1Answer: C linear equationsA linear equation with only one variable
has only one type of variable (such as q or x), even if it appears
more than once. 3x 4 = 7 + 2xA one-variable equation, when it is
graphed, makes either a horizontal or vertical line on the
graph.
Solving Linear Equations with UndoUndo parentheses by
simplifying and distributingMultiply and divide constants and
termsCombine like termsUse inverse operations to undo addition and
subtractionUse inverse operations to undo multiplication and
divisionDistributing: How to do and Why it works.3(2+4)= 183(6) =
1818 = 18
3(2 +x) = 183(2)+3x = 186 + 3x = 186 +3x -6 = 18-63x = 123x /3 =
12/3x=4
3(2+4) = 183(2) + 3(4)=186 + 12 = 18 18 = 18
Undo Parentheses by Simplifying and DistributingTo undo the
parentheses, simplify the amount in the parentheses if possible. If
you pay 8.5% tax on a $12 shirt and $23 pants, youre paying the
same tax whether you buy them together or separately. 0.085(12 +23)
= 0.085(12) + 0.085(23)The distributive property is true because
multiplication is actually repeated addition.3(2+4) =(2+4) + (2+4)
+ (2+4) = 2 + 4 + 2 + 4 + 2 + 4 (2 + 2 + 2) + (4 +4 +4)=3(2) + 3
(4) =
Kata earns $250 for every website she designs plus a bonus of 3%
of $55 for each referral from her websites. Last month, she also
earned a $130 month-end bonus. Kata had three referrals from each
of her websites and earned $3,699.30. How many websites did she
design? Using UnPac, you can write a linear equation with one
variable to solve the problem. Let w=the number of websites:250w +
0.03(55*3w) + 130 = 3699.3Undo the parentheses250w + 0.03(55*3w) +
130 = 3699.3250w + 0.03(165w) + 130 = 3699.3Now multiply 0.03 times
165 to distribute it and remove the parentheses.250w + 4.95w+ 130 =
3699.3
Multiply or Divide Constants and TermsRemember, a term is a
constant, with or without variables, or just a variable, such as:
5x 420 x 2xy.The constant in a term with a variable is called the
coefficient.
Youve already multiplied a constant and a term by multiplying 55
by 3w, which equals 165w.There are no more terms to multiply or
divide in the example equation:250w + 4.95w+ 130 = 3699.3
Combine Like TermsTo combine like terms, add or subtract terms
with the same variable by adding or subtracting the coefficients.Do
you see any terms with the same variable in our sample problem?250w
+ 4.95w+ 130 = 3699.3So, add the coefficients of w. (250
+4.95)254.95w + 130 = 3699.3
Undo Addition and Subtraction254.95w + 130 = 3699.3 -130
-130254.95w = 3569.3Undo Multiplication and Division254.95w =
3569.3254.95 254.95 w = 14Thinking AlgebraicallyApplications of
InequalitiesHave you ever?Saved money to pay for a vacation or
car?Read a manual to find the most weight a truck could
hold?Figured out the most you could spend on clothes?
Often, a solution is not limited to a single value, but consists
of a range of values. You might need to spend less than a certain
amount on groceries, keep your load to less than the maximum your
truck can carry, or save at least enough for that trip to Hawaii.
You can use inequalities to solve these types of problems. Unlike
equations, inequalities have a range of possible solutions. The
solution is a range of values that makes the inequality true. The
maximum weight allowed in an elevator is usually 3,400 pounds, but
does that mean it is the only weight allowed? Of course not.
Anything below or equal to 3,400 pounds is allowed. The inequality
sign to describe this limit is the less than or equal to sign ().
weight allowed 3400 poundsThe weight allowed on the elevator is
less than or equal to 3,400 pounds.Instead of an equal sign =, an
inequality uses one of the following signs: > is greater than is
greater than or equal to < is less than is greater than or equal
to
You can show the range of values of an inequality by graphing it
on the number line.
Even though inequalities have more than one solution, you can
still solve them by using undo. There is one big difference. When
multiplying or dividing both sides by a negative number, flip the
direction of the inequality sign. Why? Examine this
inequality.-2< 3-2 -1< or > 3-1False 2< -3True 2>
-3
UnderstandA stock started out at $8.24 per share. The next two
days, it went down by the same amount each day. Elias wants to buy
the stock if its less than or equal to $5 per share. What drop in
price for two days in a row would cause the stock to be $5 per
share or less? Graph the answer on a number line.What inequality
sign would you use in this problem?
PlanA stock started out at $8.24 per share. The next two days,
it went down by the same amount each day. Elias wants to buy the
stock if its less than or equal to $5 per share. What drop in price
for two days in a row would cause the stock to be $5 per share or
less? Graph the answer on a number line.Write an inequality to
represent the problem. 8.24 - 2x 5Attack 8.24 - 2x 5To undo an
inequality:Undo parentheses by simplifying whats in the parentheses
and distributing multiplication.Multiply and divide constants with
terms.Combine like terms.Use inverse operations to undo addition
and subtraction.Use inverse operations to undo multiplication and
division. Determine whether to flip the inequality sign.Solve the
inequality and graph it on a number line.
AttackUse inverse operations to undo addition and
subtraction.8.24 - 2x 5-8.24 -8.24 - 2x -3.24Use inverse operations
to undo multiplication and division. Determine whether to flip the
inequality sign. - 2x -3.24 -2 -2 x 1.62
Graph x 1.62
To graph a line that represents or , make a solid dot at the
endpoint on the number line. If the symbol is < or > , then
make an open dot to indicate that the endpoint is not a
solution.
CheckTo check an inequality, first write it as an equation. Then
use the substitution process to check your answer. Change the
inequality sign to an equal sign.
8.24 2 (1.62) = 58.24 3.24 = 5 5 = 5
