Math 2131 Course Description

1
Math 2131 Course Description

• An exploration of basic concepts and operations
  of rational numbers and generalizations to
  algebra. Problem Solving.

2
Day 1 Agenda

• Personal introductions
• Syllabus
• Introduction to the course web site
• Introduction to problem solving strategies

3
Problem Solving

• Polyas advice
• Understand the problem
• Devise a plan (strategy)
• Carry out the plan
• Look back

4
Initial Problem (p. 5)

• Arrange the whole numbers from 1 to 6 in the
  circles so that the sum of each side is 12.

5
Commonly employed strategies

• Guess and test
• Use a variable
• Draw a picture
• Look for a pattern
• Make a list
• Solve a simpler prob.
• Draw a diagram.
• Use direct reasoning
• Use indirect reasoning
• Use properties of numbers
• Solve an equivalent problem

• Work backward
• Use cases
• Solve an equation
• Look for a formula
• Do a simulation
• Use a model
• Use dimensional analysis
• Identify subgoals
• Use coordinates
• Use symmetry

6
A Second Problem (p. 3)

• Lins rectangular garden has an area of 782
  square feet. The length of the garden is 5 less
  than three times its width. What are the
  dimensions of Lins garden?

7
Problem 3 Cryptarithm (p. 7)

• Using all of the digits 0, 1, 2, 3, 6, 7, and 9
  in place of the letters where no letter
  represents two different digits, determine the
  value of each letter.
• s u n
• f u n
• s w i m

8
Consider guess test if

• limited number of possibilities involved
• need to develop a better understanding of problem
• cases can be attempted systematically
• choices have been narrowed using other strategies
• no clue how else to proceed

9
Problem 4 (p. 8)

• What is the greatest number that evenly divides
  the sum of any three consecutive whole numbers?

10
Problem 5 (p. 9)

• It takes 64 cubes to fill a cubical box that has
  no top. How many cubes are not touching a side
  or the bottom?

11
Now entering Section 1.2
12
Strategy 4 Look for Pattern

• How many different downward paths are there from
  A to B in the grid above? A path must travel on
  the lines.

13
Sequences

• A sequence is an ordered arrangement of numbers.
  Examples
• Even counting numbers 2, 4, 6, 8, 10, . . . .
• Odd counting numbers 1, 3, 5, 7, . . . .
• Square counting numbers 1, 4, 9, 16, . . . .
• Powers of three 1, 3, 9, 27, 81, . . . .
• Fibonacci numbers 1, 1, 2, 3, 5, 8, . . . .

14
Indicators for pattern strategy

• Given a list of data
• A sequence of numbers is involved
• Listing special cases helps address complex
  problems
• Asked to make a prediction or generalization
• Information can be expressed in organized manner

15
Strategy Make a List

• The number 10 can be expressed as the sum of four
  odd numbers in three ways
• 1). 10 7 1 1 1
• 2). 10 5 3 1 1
• 3). 10 3 3 3 1
• In how many ways can 20 be expressed as the sum
  of eight odd numbers?

16
Indicators of list strategy

• See clues on page 20 of text

17
Strategy Solve a simpler prob.

• Evaluate the sum

18
Indicators for this strategy

• See clues on page 23 of text.