MA201
Dr. Byrne Fall 2011
Chapter 1:
Problem Solving
Problem Solving
Polya’s 4-step process:
1.
2.
3.
4.
Understand the problem.
Devise a plan.
Carry out the plan.
Look back.
Problem Solving
Polya’s 4-step process:
1.
2.
3.
4.
Understand the problem.
Devise a plan.
Carry out the plan.
Look back.
Understand
Problem Solving
Polya’s 4-step process:
1.
2.
3.
4.
Understand the problem.
Devise a plan.
Think
Carry out the plan.
Look back.
Understand
Problem Solving
Polya’s 4-step process:
1.
2.
3.
4.
Understand
Understand the problem.
Devise a plan.
Think
Carry out the plan.
Do
Look back.
Problem Solving
Polya’s 4-step process:
1.
2.
3.
4.
Understand
Understand the problem.
Devise a plan.
Think
Carry out the plan.
Do
Look back.
Check
Problem Solving
Polya’s 4-step process:
1.
2.
3.
4.
Understand
Understand the problem.
Devise a plan.
Think
Carry out the plan.
Do
Look back.
Check
Compare: NCSS Problem Solving Steps
1.
2.
3.
4.
5.
Identify the problem.
Get the facts related to the problem.
Create a plan.
Carry out the plan.
Check your plan and adjust if needed.
Polya’s 4-step process:
1.
2.
3.
4.
Understand the problem.
Devise a plan.
Carry out the plan.
NCSS Problem Solving Steps
Look back.
•
•
•
•
•
Identify the problem.
Get the facts related to the
problem.
Create a plan.
Carry out the plan.
Check the plan and adjust if
needed.
Class Activity for Ch 1:
Instructions and Example
•
For a given word problem: (These instructions will
be repeated.)
a) Solve the problem with your group.
b) Consider how you applied Polya’s 4 steps.
c) If ‘trial and error’ was the strategy, is there another way you
could have solved the problem?
d) Include a careful diagram that you would provide to aid
elementary students solving this problem.
e) Compare your group’s solution with that of the other group
working on the same question.
f) Prepare a short 5 minute presentation comparing and
contrasting the way groups worked on the problems and
whether Polya’s 4 steps were useful for either group.
In Class Example
HW: Section 1.1 / Problem Set A
1. a. If the diagonals of a square are drawn
in, how many triangles of all sizes are
formed?
b. Describe how Polya’s four steps were used
to solve part a.
Step 1: Understand the problem.
If the diagonals of a square are drawn in,
how many triangles of all sizes are
formed?
• Draw a picture of a square with the diagonals drawn in.
• The question asks, “how many triangles?”
• “CLUE”: how many triangles of all sizes ?
Step 1: Understand the problem.
If the diagonals of a square are drawn in,
how many triangles of all sizes are
formed?
• Draw a picture of a square with the diagonals drawn in.
• The question asks, “how many triangles?”
• “CLUE”: how many triangles of all sizes ?
Step 1: Understand the problem.
If the diagonals of a square are drawn in,
how many triangles of all sizes are
formed?
• Draw a picture of a square with the diagonals drawn in.
• The question asks, “how many triangles?”
• “CLUE”: how many triangles of all sizes ?
Step 1: Understand the problem.
If the diagonals of a square are drawn in,
how many triangles of all sizes are
formed?
• Draw a picture of a square with the diagonals drawn in.
• The question asks, “how many triangles?”
• “CLUE”: how many triangles of all sizes ?
Step 1: Understand the problem.
If the diagonals of a square are drawn in,
how many triangles of all sizes are
formed?
• Draw a picture of a square with the diagonals drawn in.
• The question asks, “how many triangles?”
• “CLUE”: how many triangles of all sizes ?
Step 2: Devise a plan.
How to count the # of triangles of all sizes?
Plan: Figure out how many sizes there are
and count them from smallest to largest.
Step 3: Carry out plan.
First, how many sizes of triangles are there?
Step 3: Carry out plan.
First, how many sizes of triangles are there?
Step 3: Carry out plan.
First, how many sizes of triangles are there?
‘little’
‘big’
Step 3: Carry out plan.
Next, count how many of each size:
# little?
# big?
Step 4: Look back.
We’ve counted 8 triangles:
4
4
If the diagonals of a square are drawn in, how
many triangles of all sizes are formed?
Answer: 8
Step 4: Look back.
We’ve counted 8 triangles:
4
4
If the diagonals of a square are drawn in, how
many triangles of all sizes are formed?
Answer: 8
Problem Solving – In Class
You will be assigned a chapter problem in small working groups.
1.
Small groups: Solve the problem as a group. As a group, consider how
you applied Polya’s 4 steps by filling out the sheet.
2.
Individually: When the time is announced, spend one minute individually
drawing a diagram, table or manipulative that could be used to explain the
problem to a class of elementary students.
3.
Combined groups: Compare your group’s solution with that of the other
group working on the same question. Prepare a short 5 minute
presentation comparing and contrasting the way groups worked on the
problems and whether Polya’s 4 steps were useful for either group.
-- pick one person to describe the problem and draw a diagram
-- pick one person to describe the solution (1 minute only)
-- pick one person to discuss the effectiveness of Polya’s 4 steps
Problem Solving – Homework
Choose 3 problems from Section 1.1 that were not
assigned in class.
Solve the problem using Polya’s 4 steps, clearly
dividing the steps of your solution and showing
all your work.
Carefully make a diagram that you could use in a
class to aid elementary students that were
assigned this problem.