6th Grade Math Textbook, Fundamentals

xiii

Practically every moment of our lives, we are faced with decisions about what
to do and how to do it. Sometimes these decisions require a lot of thought;
other times, they are instinctive reactions. But in almost all cases, they can be
considered to be problem-solving experiences.

Mastery in problem solving depends on critical thinking. To think critically, you
must be able to organize your thoughts. The problem-solving steps outlined
above will help you do just that.
With this example, you can see the power that
some problem-solving strategies can have.
Our objective in this book is to introduce you
to ten problem-solving strategies that will be
invaluable not only when you work with
mathematics, but also in other situations,
both in and out of school. One caution:
Just reading these problem-solving strategy
sections diligently will not guarantee that
these methods will become part of your
regular thought processes. You must apply
these problem-solving techniques as often as
you can so that they dobecome part of your
regular thought processes.

Problem:Your school principal asks you

to organize a single elimination basketball

tournament, one in which one loss

eliminates a team. If 25 teams enter the

tournament, how many games have to be

played in order to have a winner?

Solution:A typical solution would require

modeling the tournament and then counting

how many games are actually played until

there is one winner. However, using a

problem-solving strategy called Adopt a

Different Point of View, you can answer this

problem instantly. Instead of modeling the

tournament and looking at the winners of

each game as the teams progress to a

championship, consider the losers. How

many losers must there be to have a

champion among 25 teams? Clearly, with

one winner, there must be 24 losers. The

number of games necessary to have 24

losers is 24. Problem solved!

Consider the following problem.

Read to understand what

is being asked.

Select a strategy.

Apply the strategy.

Check to make sure your

answer makes sense.

Problem-Solving Steps

1.Guess and Test, p. 24

2.Organize Data, p. 48

3.Find a Pattern, p. 66

4.Make a Drawing, p. 142

5.Solve a Simpler Problem, p. 168

6.Reason Logically, p. 202

7. Adopt a Different Point of View, p. 266
   8.Account for All Possibilities, p. 296
   9.Work Backward, p. 324
   10.Consider Extreme Cases, p. 376

Problem-Solving Strategies