The document discusses translating verbal phrases into algebraic expressions and using verbal models to write mathematical equations and inequalities. It provides examples of common verbal phrases involving numbers, operations, and variables and their corresponding algebraic translations. It also outlines a three-step process for writing a mathematical model from a word problem: 1) write a verbal model, 2) assign labels, and 3) write the algebraic model. Finally, it demonstrates this process with a sample word problem about the cost of dim sum plates after tax.

1. TRANSLATING VERBAL PHRASES
Verbal Phrase Expression
The sum of six and a number 6 + xsum +six 6number x
Eight more than a number y + 8more than +
A number plus five n + 5plus +
A number increased by seven x + 7+increased
A number decreased by nine n – 9–decreased
Ten times a number
Seven divided by a number
10 • n
7
x
times
divided
•
The sum of six and a number 6 + x
Eight numberEight more than a number
fivenumberA number plus five
sevennumberA number increased by seven
ninenumberA number decreased by nine
Ten numberTen times a number
8yy + 8
1010 • nn
5nn + 5
7xx + 7
9nn – 9
or ( )10n10n( )n(10 )n10(10n)
Seven 7
x
number

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.
Expressions
Phrases
Equations or
Inequalities
Sentences

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

4. USING A VERBAL MODEL
Use three steps to write a mathematical model.
WRITE A
VERBAL MODEL.
ASSIGN
LABELS.
WRITE AN
ALGEBRAIC MODEL.
Writing algebraic expressions, equations, or inequalities
that represent real-life situations is called modeling.
The expression, equation, or inequality is a
mathematical 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.
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.
SOLUTION

6. LABELS
VERBA
L
MODEL
Writing an Algebraic Model
Cost per
plate •
Number of
plates = Bill Tax–
Cost per plate = 2
Number of plates = p
Amount of bill = 25.20
Tax = 1.20
(dollars)
(dollars)
(dollars)
(plates)
25.20 1.20–2 =p
2p = 24.00
p = 12
Your group ordered 12 plates of food costing $24.00.
ALGEBRAIC
MODEL

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

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?
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?
b.

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

10. Using a Verbal Model
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.
Is it reasonable for the pilot to fly directly to Chicago at 300
miles per hour or must the pilot take some other action?
b.