Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Math 4010-002Lecture 2: Problem Solving

Ben Trahan

Department of MathematicsUniversity of Utah

January 14, 2010

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Outline

1 The Pool Problem

2 The Problem Solving ProcessPolya’s Four Steps

3 Some Problem Solving MethodsGuess and checkDraw a PictureAssign a Variable

4 One Last Problem

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Pool Border Problem

How many 1m × 1m square tiles does it take to make aborder around a square pool?

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

One Solution

You can lay L tiles across thetop, bottom, and each side.You need one more for eachcorner. Total, that is:

L + L + L + L + 4

I can represent this solution asa diagram.

L

L

LL

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Some Options

Here are a few options:

1 Draw a picture

2 Consider special cases, then generalize

3 Look for a formula

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Group Work

Take a few minutes to pictorially explain how a student mightcome up with each of these answers:

1 4(n + 1)

2 4n + 4

3 2n + (2n + 2)

4 4(n + 2) − 4

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Outline

1 The Pool Problem

2 The Problem Solving ProcessPolya’s Four Steps

3 Some Problem Solving MethodsGuess and checkDraw a PictureAssign a Variable

4 One Last Problem

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Problems versus Exercises

Exercises require the mechanical application of a process

Problems require creativity and deep thought

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Polya’s Four Steps

1 Understand the Problem

2 Devise a Plan

3 Carry Out the Plan

4 Look Back

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

The Pool Problem Example

1 Understand the Problem

What might trip you up?What information is relevant?

2 Devise a Plan

Solve an example, then generalize

3 Carry Out the Plan

Assign a variable

4 Look Back

Is my answer reasonable?Does the formula work for this example?

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Outline

1 The Pool Problem

2 The Problem Solving ProcessPolya’s Four Steps

3 Some Problem Solving MethodsGuess and checkDraw a PictureAssign a Variable

4 One Last Problem

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Guess and Check

The Indian mathematician Ramanujan observed that thenumber 1729 was very interesting, because it was the smallestcounting number that could be expressed as a sum of a pair ofcubes in two different ways. Find the pairs of cubes that addup to 1729.

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Polya’s Four Steps

The Indian mathematician Ramanujan observed that thenumber 1729 was very interesting, because it was the smallestcounting number that could be expressed as a sum of a pair ofcubes in two different ways. Find the pairs of cubes that addup to 1729.

Understand the Problem: What are we trying to find?

Devise a Plan: How can we organize our guessing?

Carry Out the Plan: Grind it out. If it’s taking too long, find anew plan!

Look Back: Is our answer correct?

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

A Table of Cubes

x x3

1 12 83 274 645 1256 2167 3438 5129 729

10 100011 133112 1728

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Triangular Numbers

A triangular number is a whole number that can berepresented by an array of dots in a particular shape. The firstfour triangular numbers are

What is the sixth triangular number?

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Harder Questions

What is the 10th triangular number?

What is the nth triangular number?

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Solution

There are 5 × 4 dots in two copies of the triangle. There are5×4

2 = 10 dots in the original triangle.

So what is the 10th triangular number? What is the nth?

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Another Problem Solving Outline

1 Understand the problem

2 Translate the problem into math

3 Solve the mathematical problem

4 Interpret the answer

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Two Lakes

The surface of Big Lake is 31 feet above the surface of LongLake. Long lake is half as deep as Big Lake and the bottom ofLong Lake is 8 feet below the bottom of Big Lake. How deep iseach lake?

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Two Lakes Solution

The surface of Big Lake is 31 feet above the surface of LongLake. Long lake is half as deep as Big Lake and the bottom ofLong Lake is 8 feet below the bottom of Big Lake. How deep iseach lake?

Say the depth of Big Lake is d . We know:

Depth of Long Lake =1

2d

Height of Long Lake = Depth of Long Lake =1

2d

Height of Big Lake = Depth of Big Lake + 8 = d + 8

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Two Lakes Solution

Height of Big Lake = 31 + Height of Long Lake

d + 8 = 31 +1

2d

2d + 16 = 62 + d

d + 16 = 62

d = 46

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Outline

1 The Pool Problem

2 The Problem Solving ProcessPolya’s Four Steps

3 Some Problem Solving MethodsGuess and checkDraw a PictureAssign a Variable

4 One Last Problem

Math4010-002

Ben Trahan

The PoolProblem

The ProblemSolvingProcess

Polya’s FourSteps

Some ProblemSolvingMethods

Guess and check

Draw a Picture

Assign a Variable

One LastProblem

Points on a Circle

If six points are placed on a circle, and each pair of points isjoined by a segment, how many segments are there?

If n points are placed on a circle, how many segments arethere?