Transcript Step 2 Devise a plan to solve the problem.

Math in Our World
Section 1.3
Problem Solving
Learning Objectives
State the four steps in the basic problem solving
procedure.
Solve problems by using a diagram.
Solve problems by using trial and error.
Solve problems involving money.
Solve problems by using calculation.
Polya’s Four-Step Problem-Solving
Procedure
Step 1 Understand the problem.
Read the problem slowly, jotting down the key ideas
Step 2 Devise a plan to solve the problem.
Draw a diagram, find a formula, look for patterns
Step 3 Carry out the plan to solve the problem.
Solve the problem, follow the numbers,
create an equation
Step 4 Check the answer.
Does your answer make sense?
Did you solve for the requested unknown?
EXAMPLE 1 Solving a Problem by Using a
Diagram
A gardener is asked to plant eight tomato plants
that are 18 inches tall in a straight line with 2
feet between each plant. How much space is
needed between the first plant and the last one?
Be careful—what seems like an obvious solution is not
always correct! You might be tempted to just multiply 8
by 2, but instead we will use Polya’s method.
EXAMPLE 1
Solving a Problem by
Using a Diagram
SOLUTION
•Step 1 Understand the problem. In this case, the key information
given is that there will be eight plants in a line, with 2 feet between
each. We are asked to find the total distance from the first to the last.
•Step 2 Devise a plan to solve the problem. When a situation is
described that you can draw a picture of, it’s often helpful to do so.
•Step 3 Carry out the plan to solve the problem. The figure would
look like this:
Now we can use the picture to add up the distances:
2 + 2 + 2 + 2 + 2 + 2 + 2 = 14 feet
•Step 4 Check the answer. There are eight plants, but only seven
spaces of 2 feet between them. So 7 x 2 = 14 feet is right.
EXAMPLE 2
Solving a Problem Using
Trial and Error
Suppose that you have 10 coins consisting of
quarters and dimes. If you have a total of $1.90,
find the number of each type of coin.
EXAMPLE 2
Solving a Problem Using
Trial and Error
SOLUTION
•Step 1 Understand the problem. We’re told that we have a
total of 10 coins and that some are dimes (worth $0.10 each) and
the rest are quarters (worth $0.25 each). The total value is $1.90.
The problem is to find how many quarters and dimes together are
worth $1.90.
•Step 2 Devise a plan to solve the problem. One strategy that
can be used is to make an organized list of possible
combinations of 10 total quarters and dimes and see if the sum is
$1.90. For example, you may try one quarter and nine dimes.
This gives 1 x $0.25 + 9 x $0.10 = $1.15.
EXAMPLE 2
Solving a Problem Using
Trial and Error
SOLUTION
•Step 3 Carry out the plan. Since one quarter and nine dimes is
wrong, try two quarters and eight dimes.
This doesn’t work either.
So continue…
Answer: six quarters and four dimes.
•Step 4 Check the answer. In this case, our answer can be checked
by working out the amounts: 6 x $0.25 + 4 x $0.10 = $1.90.
EXAMPLE 3
Solving a Problem
Involving Salary
So you’ve graduated from college and you’re
ready for that first real job. In fact, you have two
offers! One pays an hourly wage of $19.20 per
hour, with a 40-hour work week. You work for 50
weeks and get 2 weeks’ paid vacation. The
second offer is a salaried position, offering
$41,000 per year. Which job will pay more?
EXAMPLE 3
Solving a Problem
Involving Salary
SOLUTION
•Step 1 Understand the problem. The important information is…
– The hourly job pays $19.20 per hour for 40 hours each week
– You will be paid for 52 weeks per year.
– We are asked to decide if that will work out to be more or less
than $41,000 per year.
(The fact that you get 2 weeks’ paid vacation is irrelevant to the
problem.)
•Step 2 Devise a plan to solve the problem. We can use
multiplication to figure out how much you would be paid each week
and then multiply again to get the yearly amount. Then we can
compare to the salaried position.
EXAMPLE 3
Solving a Problem
Involving Salary
SOLUTION
•Step 3 Carry out the plan to solve the problem.
Multiply the hourly wage by 40 hours;
$19.20 x 40 = $768
This shows that the weekly earnings will be $768.
Now we multiply by 52 weeks:
$768 x 52 = $39,936
This gives an annual income of $39,936.
The salaried position, at $41,000 per year, pays
more.
•Step 4 Check the answer. We can figure out the hourly wage of the
job that pays $41,000 per year. We divide by 52 to get a weekly
salary of $788.46. Then we divide by 40 to get an hourly wage of
$19.71. Again, this job pays more.
EXAMPLE 4
Solving a Problem by
Using Calculation
A 150-pound person walking briskly for 1 mile
can burn about 100 calories. How many miles
per day would the person have to walk to lose 1
pound in one week? It is necessary to burn
3,500 calories to lose 1 pound.
EXAMPLE 4
Solving a Problem by
Using Calculation
SOLUTION
•Step 1 Understand the problem.
– A 150-pound person burns 100 calories per 1 mile
– The person needs to burn 3,500 calories in 7 days to lose 1
pound
– The problem asks how many miles per day the person has to
walk to lose 1 pound in 1 week.
•Step 2 Devise a plan to solve the problem. We will calculate
how many calories need to be burned per day and then divide by
100 to see how many miles need to be walked.
EXAMPLE 4
Solving a Problem by
Using Calculation
SOLUTION
•Step 3 Carry out the plan. Since 3,500 calories need to be
expended in 7 days, divide to find out how many are needed to be
burned per day.
3,500 ÷ 7 = 500
Then divide 500 by 100 to get 5 miles. It is necessary to walk
briskly 5 miles per day to lose 1 pound in a week.
•Step 4 Check the answer. Multiply 5 miles per day by 100
calories per mile by 7 days to get 3,500 calories.
Classwork
p. 32-34: 5, 10, 13, 15, 19, 22, 28, 30, 33, 37, 41, 46