Problem-Solving Strategies in Mathematics

From Common Approaches to Exemplary Strategies

Alfred S Posamentier, Stephen Krulik

• Editore: WSPC
• Anno: 2015
• ISBN:
  9789814651653
• Argomento: Mathematics
• Categoria: Mathematics in Education

This book introduces ten problem-solving strategies by first presenting the strategy and then applying it to problems in elementary mathematics. In doing so, first the common approach is shown, and then a more elegant strategy is provided. Elementary mathematics is used so that the reader can focus on the strategy and not be distracted by some more sophisticated mathematics.

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Contents:

• Logical Reasoning
• Pattern Recognition
• Working Backwards
• Adopting a Different Point of View
• Considering Extreme Cases
• Solving a Simpler Analgous Problem
• Organizing Data
• Making a Drawing or Visual Representation
• Accounting for All Possibilities
• Intelligent Guessing and Testing

Readership: Undergraduates and general public interested in Mathematics.
Key Features:

• Each author has more than 50 years of experience in the field
• Problem solving in mathematics is the key element for success in the field
• The strategies presented can be applied to many other applications — beyond mathematics

Chapter 1

Logical Reasoning

It must seem rather redundant to have a chapter entirely devoted to a strategy referred to as logical reasoning. Regardless of the strategy being used to solve a problem, it would seem that some logical thinking must permeate the use of all the strategies. After all, to many people, problem solving is almost synonymous with logical reasoning, or logical thought. So, why have this chapter and why isolate this strategy?

In everyday life, we rely on logical reasoning when arguing a point with someone. After all, when we are having any kind of debate, we expect certain arguments to generate specific responses. At work, you can use a logical chain of arguments to change the way something is done in the office. We use logical reasoning to generate a chain of statements that, we hope, will lead to the conclusion we desire. In a courtroom, for instance, lawyers use logical reasoning to make their case for a desired verdict. If we are to meet someone in two days and today is Saturday, logic tells us we are meeting him, or her, on Monday.

In problem solving in mathematics, there are some problems that essentially do not involve any of the other strategies we normally use — some of which are presented in this book. Instead, they require us to arrive at a conclusion, which is reached by careful thought, and making a series of statements that follow one another in a logical chain of reasoning. For example, let us look at the following problem.

Find all pairs of prime numbers whose sum is 741.

Many people would make a list of all of the prime numbers less than 741 and search for those pairs that add up to 741. However, we can simplify our work with some logical reasoning. If the sum of two numbers is an odd number, one of the addends must be odd, the other must be even. But there is only one even prime number, namely 2. Therefore, the other number must be 739 (and 739 is a prime number). We have found all pairs that meet the given requirements.

Let us consider another problem where logical reasoning enables us to solve it.

A palindromic number is one that reads the same forwards or backwards. Some examples of 3-digit and 4-digit palindromes are 373, and 8668. Maria wrote all the 3-digit palindromes on slips of paper and put them into a large box. Miguel wrote all the 4-digit palindromes on slips of paper and put them into the same box. The teacher stirred them all up, mixed them well, and asked Laura to pull one slip from the box without looking. What is the probability that she chose a 4-digit palindrome?

One method would be to write out all the 3-digit and 4-digit palindromes, count them all, and figure out the requested probability. This would work, even if it were somewhat time consuming. But if we use our logical reasoning strategy, we can simplify our work as follows. One example of a 3-digit palindrome might be 373. To make it a 4-digit palindrome all we have to do is repeat the middle digit, to obtain 3773. In fact, we can make every 3-digit palindrome into a 4-digit palindrome by simply repeating the middle digit once. Thus, the number of 4-digit palindromes is the same as the number of 3-digit palindromes, and so the probability of picking a 4-digit palindrome is one out of two, or

.

We will consider another example of how simple logical reasoning makes the solution of a problem rather simple.

On a shelf in the florist’s store, there are three boxes of ornamental bows to put on gift wrapped boxes. Mark went to put the three labels—“Red”, “White”, and “Mixed” (red and white) on the boxes. Unfortunately, he put the labels back, but put all three on the wrong boxes. Because the boxes are on a high shelf, Mark cannot look into the boxes. He knows all three are mislabeled, and he wants to reach up and pick one bow from one of the boxes. From which box should he pick the one bow in order to label all three boxes correctly?

Let us do some logical reasoning here. First, notice that whatever we say about the box labeled “White” we can also say about the box labeled “Red”. There is a kind of symmetry there. So, let Mark pick the single bow from the box marked “Mixed”. If it is red, then he knows that this box is really the box containing only red bows, since it cannot be the “Mixed” box. Label it “Red”. The box labeled “White” cannot be all white, so it must be the “Mixed” box. Finally, the box incorrectly labeled “Red” must be the white box.

Notice that each of these problems requires not much more than some logical reasoning and careful thinking to reach a solution. This is not to say that we do not require logical thinking when using the other problem-solving strategies; however, the problems presented in this chapter rely almost exclusively on logical reasoning to reach a solution effectively.

PROBLEM 1.1

Max begins counting the natural numbers forward as 1, 2, 3, 4, . . . , while Sam is counting at the same speed but in the opposite direction—counting backwards from the number x as follows x, x – 1, x – 2, x – 3, x – 4, . . . . When Max says the number 52, Sam says the number 74. With which number (x) did Sam start with in his counting backwards procedure?

A Common Approach

Faced with this problem most people will probably try to simulate the situation being described; that is, carrying both accounting procedures simultaneously to see what would result. The difficulty is that since one would not know where to begin the backward counting, the forward counting—in a trial and error procedure—would most likely be employed. This would not only be confusing, but very difficult to carry out.

An Exemplary Solution

Here, we will employ some logical reasoning. As Max counts 52 numbers, Sam will also be counting 52 numbers. We can designate Sam’s 52nd number as x–51. However, we know that this number is to be 74. Therefore, we can equate them as x – 51 = 74, and then x = 125.

PROBLEM 1.2

We have 100 kg of berries and water, where 99% of the weight is water. A while later the water content of the mixture is 98% water. How much do the berries weigh?

A Common Approach

A common wrong answer is that with an evaporation of 1% water, that 99% must be the berries, which would imply that the berries weigh 99 kg. This is wrong!

An Exemplary Solution

Here we will have to use some logical reasoning to ascertain what is required. Initially, the mixture is 99% water, meaning that it contains 99 kg of water and 1 kg of dry matter, or 1% of the berries’ mass. The mass of the dry matter does not change: at the end of the drying process, its weight remains 1 kg. In the meantime, however, the proportion of the total mass that is not water has doubled, to 2%.

In order for something that has a fixed quantity (our 1 kg of dry ...