The sum of six and a number

1
The sum of six and a number
6 x
6 x
number
x

sum
six
6
Eight more than a number
y 8
y 8
8
y
A number plus five
n 5
n 5
5
n
A number increased by seven
x 7
x 7
7
x
n 9
A number decreased by nine
n 9
9
n
(10n)
(10 )
Ten times a number
10 n
10 n
10n
( )
or
10
n
n
10
n
Seven divided by a number
7
x
2
USING A VERBAL MODEL
In Mathematics there is a difference between a
phrase and a sentence. Phrases translate into
expressions sentences translate into equations
or inequalities.
Phrases
Expressions
Equations or Inequalities
Sentences
3
USING A VERBAL MODEL
Phrase
Expression
The sum of six and a number
6 x
The sum of six and a number is
6 x
The sum of six and a number is twelve.
6 x 12
The sum of six and a number is less than twelve.
6 x lt 12
4
USING A VERBAL MODEL
Writing algebraic expressions, equations, or
inequalities that represent real-life situations
is called modeling.
The expression, equation, or inequality is a
mathematical model.
Use three steps to write a mathematical model.
WRITE A VERBAL MODEL.
ASSIGN LABELS.
WRITE AN ALGEBRAIC MODEL.
5
You and three friends are having a dim sum lunch
at a Chinese restaurant that charges 2 per
plate. You order lots of plates. The waiter
gives you a bill for 25.20, which includes tax
of 1.20. Use mental math to solve the equation
for how many plates your group ordered.
SOLUTION
Understand the problem situation before you
begin. For example, notice that tax is added
after the total cost of the dim sum plates is
figured.
6
VERBAL MODEL



Cost per plate
(dollars)
Number of plates
(plates)
Amount of bill
(dollars)
Tax
(dollars)

2p
24.00
p

12
7
A PROBLEM SOLVING PLAN USING MODELS
Ask yourself what you need to know to solve the
problem. Then write a verbal model that will give
you what you need to know.
Ask yourself what you need to know to solve the
problem. Then write a verbal model that will give
you what you need to know.
VERBAL MODEL
Assign labels to each part of your verbal problem.
Assign labels to each part of your verbal problem.
Use the labels to write an algebraic model based
on your verbal model.
Use the labels to write an algebraic model based
on your verbal model.
Solve the algebraic model and answer the original
question.
Solve the algebraic model and answer the original
question.
Check that your answer is reasonable.
8
JET PILOT A jet pilot is flying from Los Angeles,
CA to Chicago, IL at a speed of 500 miles per
hour. When the plane is 600 miles from Chicago,
an air traffic controller tells the pilot that it
will be 2 hours before the plane can get
clearance to land. The pilot knows the speed of
the jet must be greater then 322 miles per hour
or the jet could stall.
a. At what speed would the jet have to fly to
arrive in Chicago in 2 hours?
b.
Is it reasonable for the pilot to fly directly to
Chicago at the reduced speed from part (a) or
must the pilot take some other action?
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a. At what speed would the jet have to fly to
arrive in Chicago in 2 hours?
SOLUTION
Distance to travel
VERBAL MODEL


Speed of jet
(miles per hour)
Time
(hours)
Distance to travel
(miles)

x

300
10
b.
Is it reasonable for the pilot to fly directly to
Chicago at 300 miles per hour or must the pilot
take some other action?
It is not reasonable for the pilot to fly at 300
miles per hour, because the jet could stall. The
pilot should take some other action, such as
circling in a holding pattern, to use some of the
time.