Thinking Critically 1.1 An Introduction to Problem Solving 1.3 More Problem-Solving Strategies 1.4 Algebra as a Problem-Solving Strategy 1.5 Additional Problem-Solving Strategies 1.6 Reasoning Mathematically

1.1 An Introduction to Problem Solving

Example 1.1 Toni is thinking of a number. If you double the number and add 11, the result is 39. What number is Toni thinking of? Solution 1: Guessing and checking Guess 10: 2(10) + 11 = 20 + 11 = 31 too small Guess 20: 2(20) + 11 = 40 + 11 = 51 too large Guess 15: 2(15) + 11 = 30 + 11 = 41 a bit large Guess 14: 2(14) + 11 = 28 + 11 = 39 This checks! Toni’s number must be 14.

Example 1.1-continued Toni is thinking of a number. If you double the number and add 11, the result is 39. What number is Toni thinking of? Solution 2: Make a table and looking for a pattern We need to get to 39, jump by 2 each time. Guess 8 + 6 = 14.

Problem-Solving Strategy 1 Guess and Check Make a guess and check to see if it satisfies the demands of the problem. If it doesn’t, alter the guess appropriately and check again. When the guess finally checks, a solution has been found.

Example 1.3 Using Guess and Check In the first diagram the numbers in the big circles are found by adding the numbers in the two smaller adjacent circles. Complete the second diagram so that the same pattern holds.

Problem-Solving Strategy 2 Make an Orderly List For problems that require consideration of many possibilities, make an orderly list or a table to make sure that no possibilities are missed.

Example 1.4 Make an Orderly List How many different total scores could you make if you hit the dartboard shown with three darts?

Problem-Solving Strategy 3 Draw a Diagram Draw a diagram or picture that represents the data of the problem as accurately as possible.

Example 1.5 Draw a Diagram In a stock car race the first five finishers in some order were a Ford, a Pontiac, a Chevrolet, a Buick, and a Dodge. The Ford finished 7 seconds before the Chevrolet. The Pontiac finished 6 seconds after the Buick. The Dodge finished 8 seconds after the Buick. The Chevrolet finished 2 seconds before the Pontiac.

1.3 More Problem-Solving Strategies

Problem-Solving Strategy 4 Look for a Pattern Consider an ordered sequence of particular examples of the general situation described in the problem. Then carefully scrutinize these results, looking for a pattern that may be the key to the problem.

Example 1.6 Look for a Pattern For the following numerical sequence, fill in the blanks. 1, 4, 7, 10, 13, ___, ___ 16 19 +3 +3 +3 +3 +3 +3

Example 1.6 continued For the following numerical sequence, fill in the blanks. 19, 20, 22, 25, 29, ___, ___ 34 40 +1 +2 +3 +4 +5 +6

Example 1.6 continued For the following numerical sequence, fill in the blanks. 1, 4, 9, 16, 25, ___, ___ 36 49 +3 +5 +7 +9 +11 +13

Example 1.6 continued OR notice the pattern of perfect squares: 1, 4, 9, 16, 25, ___, ___ 36 49 12, 22, 32, 42, 52, 62, 72

Problem-Solving Strategy 5 Make a Table Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Example 1.7 Applying Make a Table a. Draw the next two diagrams to continue this sequence of dots: b. How many dots are in each figure? c. How many dots would be in the one-hundredth figure?

Example 1.7 continued a. Draw the next two diagrams to continue this sequence of dots: The arrays of dots are similar, each array has one more two-dot column than its predecessor.

Example 1.7 Applying Make a Table b. How many dots are in each figure? Count the dots in each array. 1, 3, 5, 7, 9, 11, … c. How many dots would be in the one-hundredth figure?

Example 1.7 Applying Make a Table c. How many dots would be in the one-hundredth figure? Create a table. The number of 2s added is one less than the number of the term. The one-hundredth term is 1 + 99 × 2 = 199.

Problem-Solving Strategy 6 Consider Special Cases In trying to solve a complex problem, consider a sequence of special cases. This will often show how to proceed naturally from case to case until one arrives at the case in question. Alternatively, the special cases may reveal a pattern that makes it possible to solve the problem.

1.4 Algebra as a Problem-Solving Strategy

Problem-Solving Strategy 7 Use a Variable A variable is a symbol (usually a letter) that can represent any of the numbers in some set of numbers. Often a problem requires that a number be determined. Represent the number by a variable, and use the conditions of the problem to set up an equation that can be solved to ascertain the desired number.

Example 1.9 Gauss’s Insight Find the sum of the whole numbers from 1 to 100. Understand the problem. Find 1 + 2 + 3 + … + 100. Devise a plan. Let S = 1 + 2 + 3 + … + 100. Note that we could also write S = 100 + 99 + 98 + … + 1. Add these together.

Carry out the plan. S = 1 + 2 + 3 + … + 100 S = 100 + 99 + 98 + … + 1 2S = 101 + 101 + 101 + … + 101 2S = 100 × 101

The Steps in Algebraic Reasoning

Example 1.12 Setting Up and Solving an Equation: Can I Get a C? Larry has exam scores of 59, 77, 48, and 67. What score does he need on the next exam to bring his average for all five exams to 70? Let s denote Larry’s minimum score needed on the fifth exam.

Example 1. 12 Setting Up and Solving an Equation: Can I Get a C Example 1.12 Setting Up and Solving an Equation: Can I Get a C?-continued Larry has exam scores of 59, 77, 48, and 67. What score does he need on the next exam to bring his average for all five exams to 70? Multiply each side by 5 and add the sum of the first four test scores. Subtract 251 from both sides to solve for s. Larry must hope for a 99 or 100 on the last test.

Example 1.13 Solving a Rate Problem Two years ago, it took Tom 8 hours to whitewash a fence. Last year, Huck took just 6 hours to whitewash the fence. This year, Tom and Huck have decided to work together so that they’ll have time left in the afternoon to angle for catfish. How long will the job take the two boys? Let T denote the time, in hours, that Tom and Huck need together to whitewash the fence. Tom can whitewash the fence in 8 hours, he can whitewash 1/8 of the fence per hour.

Example 1.13 Solving a Rate Problem continued In T hours, Tom will have whitewashed T/8 of the fence. Huck can whitewash the fence in 8 hours, he can whitewash 1/6 of the fence per hour. Therefore he can whitewash T/6 of the fence in T hours. Working together they boys will whitewash the entire fence when

Example 1.13 Solving a Rate Problem continued Multiply both sides by 48 gives the equivalent equation. Tom and Huck can whitewash the fence in just less than three and a half hours.

1.5 Additional Problem-Solving Strategies

Problem-Solving Strategy 8 Work Backward Start from the desired result and work backward step-by-step until the initial conditions of the problem are achieved.

Problem-Solving Strategy 9 Eliminate Possibilities Suppose you are guaranteed that a problem has a solution. Use the data of the problem to decide which outcomes are impossible. Then at least one of the possibilities not ruled out must prevail. If all but one possibility can be ruled out, then it must prevail.

Example 1.16 Eliminating Possibilities Beth, Jane, and Mitzi play on the basketball team. Their positions are forward, center, and guard. Given the following information, determine who plays each position. a. Beth and the guard bought a milk shake for Mitzi. b. Beth is not a forward.

Example 1.16 Eliminating Possibilities a. Beth and the guard bought a milk shake for Mitzi. b. Beth is not a forward. Use a table. We conclude that Mitzi plays forward, Beth plays center and Jane plays guard. Beth Jane Mitzi forward center guard X X X X X X

Problem-Solving Strategy 10 The Pigeonhole Principle If m pigeons are placed into n pigeonholes and m > n, then there must be at least two pigeons in one pigeonhole.

Example 1.17 Using the Pigeonhole Principle A student working in a tight space can barely reach a box containing 12 rock CDs and 12 classical CDs. Her position is such that she cannot see into the box. How many CDs must she select to be sure that she has at least 2 of the same type of CDs? Make an orderly list: Two rock CDs and zero classical CDs One rock CD and one classical CD Zero rock CDs and two classical CDs

Example 1.17 Using the Pigeonhole Principle Make an orderly list: Two rock CDs and zero classical CDs One rock CD and one classical CD Zero rock CDs and two classical CDs Two CDs are not enough; she might get one of each kind. But if she selects a third CD, she will end up with a third rock CD, a second rock CD, a second classical CD, or a third classical CD. In any case, she will have two CDs of the same type and the condition of the problem will be satisfied.

1.6 Reasoning Mathematically

Inductive Reasoning Inductive reasoning is drawing a conclusion based on evidence obtained from specific examples. The conclusion drawn is called a generalization. An example this disproves a statement is called a counterexample.

Problem-Solving Strategy 11 Use Inductive Reasoning Observe a property that holds in several examples. Check that the property holds in other examples. In particular, attempt to find an example where the property does not hold. If the property holds in every example, state a generalization that the property is probably true in general.

A generalization that seems to be true, but has yet to be proved, is called a conjecture. Once a conjecture is given a proof, it is called a theorem.