1. The sum of six and a number The sum of six and a number number 6 + x
TRANSLATING VERBAL PHRASES
Verbal Phrase
Expression
The sum of six and a number
The sum of six and a number
number
6 + x
6 + x
sum
six x 6 +
Eight more than a number
Eight more than a number
Eight
more than
number
y + 8
y + 8 + y 8
A number plus five
A number plus five
plus
number
five
n + 5
n + 5 + n 5
A number increased by seven
A number increased by seven
increased
x + 7
x + 7 +
number
seven x 7
A number decreased by nine
A number decreased by nine
decreased
n – 9
n – 9 –
number
nine n 9
Ten times a number
(10n)
Ten times a number
times
10 • n
10 • n
(10 )
Ten
number •
( )
10n
( )
10n 10 n or 10 n n 7 x
Seven divided by a number
divided 7
Seven
number x

2. MODEL AND SOLVE REAL-LIFE PROBLEMS
In Mathematics there is a difference between a phrase and a sentence. Phrases translate into expressions; sentences translate into equations or inequalities.
Phrases
Expressions
Sentences
Equations or Inequalities

3. The sum of six and a number The sum of six and a number 6 + x 6 + x
MODEL AND SOLVE REAL-LIFE PROBLEMS
Phrase
Expression
The sum of six and a number
The sum of six and a number
6 + x
6 + x
The sum of six and a number is
6 + x =
The sum of six and a number is is
6 + x = =
In this sentence, “is” says that
one quantity is equal to one another.
Sentence
Equation
The sum of six and a number is twelve.
6 + x = 12
The sum of six and a number is twelve.
number is twelve.
6 + x = 12
is twelve.
6 + x = 12
= 12
In this sentence, the words “is less than” indicate an inequality.
Sentence
Inequality
The sum of six and a number is less than twelve.
6 + x < 12
is less than twelve.
6 + x < 12
The sum of six and a number is less than twelve.
6 + x < 12
< 12
is less than twelve.

4. WRITE AN ALGEBRAIC MODEL.
MODEL AND SOLVE REAL-LIFE PROBLEMS
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. Writing an Algebraic Model
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. – – • = = = = Your group ordered 12 plates of food costing $24.00.
Writing an Algebraic Model
VERBAL
MODEL
Cost per
plate
Number of plates • = –
Bill
Tax
LABELS
Cost per plate = 2
(dollars)
ALGEBRAIC
MODEL p
Number of plates =
(plates)
Amount of bill =
25.20
(dollars)
Tax =
1.20
(dollars) 2 = p
25.20
1.20 – 2p =
24.00 p = 12
Your group ordered 12 plates of food costing $24.00.

7. Assign labels to each part of your verbal problem.
Writing an Algebraic Model
A PROBLEM SOLVING PLAN USING MODELS
VERBAL
MODEL
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.
LABELS
Assign labels to each part of your verbal problem.
Assign labels to each part of your verbal problem.
ALGEBRAIC
MODEL
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
Solve the algebraic model and answer the original question.
Solve the algebraic model and answer the original question.
CHECK
Check that your answer is reasonable.

8. Using a Verbal Model
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?

9. Using a Verbal Model
a. At what speed would the jet have to fly to arrive in Chicago in 2 hours?
SOLUTION
You can use the formula (rate)(time) = (distance) to write a verbal model.
VERBAL
MODEL
Speed of
jet
Distance to
travel •
Time =
LABELS
Speed of jet = x
(miles per hour)
ALGEBRAIC
MODEL
Time = 2
(hours)
Distance to travel =
600
(miles) 2 x =
600 x =
300
To arrive in 2 hours, the pilot would have to slow the jet down to 300 miles per hour.

10. Using a Verbal Model 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.