HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 1.1 Thinking Mathematically

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Objectives o Understand mathematical reasoning o Distinguish between inductive and deductive reasoning o Identify arithmetic and geometric sequences

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Thinking Mathematically Mathematical thinking is important for decisions we all make every day. Possessing the ability to think mathematically makes one a better problem solver for all occasions. Reasoning is defined by the Merriam-Webster dictionary as the drawing of inferences or conclusions through the use of statements offered as explanation or justification. We begin our discussion of reasoning with inductive and deductive reasoning.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Inductive Reasoning Inductive reasoning is a line of reasoning that arrives at a general conclusion based on the observation of specific examples. Inductive reasoning can be considered a generalization.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Inductive Reasoning Consider the following argument. In New York City, it snowed 30 inches during January 2010 and 35 inches during January Therefore, New York City will receive at least 30 inches of snow every January. Does this argument use inductive reasoning?

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Inductive Reasoning (cont.) Solution Notice that in this argument, the example is specific to January 2010 and January 2011, and then a very general conclusion is made. Obviously, the likelihood of it snowing exactly the same amount each January is not very practical, but nevertheless, the argument here is inductive.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Using Inductive Reasoning Consider the following sequence of numbers. 1, 4, 9, 16, 25,... If the number pattern continues, can you conclude what the next number will be? What about the 15 th number?

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Using Inductive Reasoning (cont.) Solution If we look closely, we can see that the numbers are simply the squares of the natural numbers: etc. Therefore, the next number in the pattern would be then and so on. That means the 15 th number in the pattern would be Notice again that we begin with a specific example and continue to a generalized answer.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Inductive Reasoning Counterexample A counterexample is a contradictory example that does not satisfy our conclusion, therefore making the argument invalid. Note that one counterexample is enough to prove that a line of reasoning is false, but one positive example is never enough to prove that it is true.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Counterexamples Consider the statement: You must have a degree in computer science to become wealthy in the computer industry. Is this a valid argument?

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Counterexamples (cont.) Solution For this example, we need to look no further than two of the most famous people in the computer industry: Steve Jobs (cofounder of Apple Computers) and Bill Gates (cofounder of Microsoft). Neither Jobs nor Gates have a degree in computer science, yet both became quite wealthy in the computer industry. Hence, we have found a counterexample to our conclusion, and the argument is invalid.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Reasoning with Sequences Identify a pattern in each of the following sequences of numbers, then use the established pattern to find the next term in the sequence. a.4, 9, 14, 19, ____ b.2, 6, 18, 54, ____ c.5, 6, 8, 11, ____

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Reasoning with Sequences (cont.) Solution When trying to identify a pattern in a sequence of numbers, there is not a set method to follow. We will introduce you to several techniques in this example. The more practice you have with these techniques, the easier it will become to identify patterns. a.When considering the sequence 4, 9, 14, 19, ____, we should try to determine whether the difference between the consecutive terms is constant (that is, if the difference is the same for each consecutive pair of terms)

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Reasoning with Sequences (cont.) or if the difference between the consecutive pairs of terms varies. We can see here that there is a common difference between each number in the sequence. The common difference is 5. Thus, the next term in the sequence will be = 24.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Reasoning with Sequences (cont.) b.The sequence 2, 6, 18, 54, __ does not have a common difference between terms as in part a. Since there is no common difference, we need another approach. Instead of a common difference, perhaps there is a common ratio between the consecutive pairs of terms.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Reasoning with Sequences (cont.) We can see that each successive number is the product of the previous number and 3. So, 2 ⋅ 3 = 6, 6 ⋅ 3 = 18, 18 ⋅ 3 = 54, etc. This means that the next term would be 54 ⋅ 3 = 162. c.For the sequence, 5, 6, 8, 11, ____, there is no common difference or common ratio between the numbers. We need to ask ourselves, What do we have to do to the first term in order to obtain the second? In this case, we have to add 1 to 5 in order to get 6.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Reasoning with Sequences (cont.) For the next term, we see that the sum is = 8. We can now see a pattern developing that we need to investigate further. We added 1 to the first term and 2 to the second term. If we continue in this manner by adding 3 to the third term, 4 to the fourth term, etc., does the pattern continue? For the next term, = 11, so indeed it does. Thus, to find the next term in the sequence, we add 4 to 11 to get 15.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Arithmetic Sequences When the common difference between any two consecutive terms in a number sequence is the same, we call this an arithmetic sequence.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Geometric Sequences When the common ratio between any two consecutive terms in a number sequence is the same, we call this a geometric sequence.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Reasoning with Patterns A certain type of human cell reproduces in the following manner: 1 cell, 4 cells, 9 cells, 16 cells. Determine the number of cells present on the next production of cells.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Reasoning with Patterns (cont.) Solution The second example we did concerning inductive reasoning asked us to consider the sequence of numbers 1, 4, 9, 16,.... Notice that the first iteration contains 1 cell and the second iteration contains cells. The third and fourth iterations contain cells and cells, respectively. So, the next iteration in the sequence will contain cells.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 Consider the given figures. To build the figures, the first figure requires 16 line segments, the second figure requires 28 line segments, and the third figure requires 40 line segments. How many line segments would be required a. in the 8 th figure? b. in the n th figure?

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 Answer: 1. a. 100 b n

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Deductive Reasoning Deductive reasoning is a process that begins with commonly accepted facts and logically arrives at a specific conclusion.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Deductive Reasoning Consider the following statement. If you are a mammal, then you have lungs. How can this statement be evaluated as a deductive argument? Solution This statement has two parts: determining if you are a mammal and whether or not you have lungs. Breaking down the individual components of the statement, we can observe the following.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Using Deductive Reasoning (cont.) a.I am a mammal (premise is true). I have lungs (conclusion is true). b.I am a mammal (premise is true). I do not have lungs (conclusion is false). Notice that, in instance b., when the conclusion is false, the argument does not seem to follow “logical” thought. There is a much more detailed discussion of deductive reasoning in Chapter 3.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Inductive versus Deductive Reasoning To illustrate the difference between inductive and deductive reasoning, consider the following process. Choosing a positive integer, multiplying it by 2, and adding 1 to the product will result in an odd number. a. Evaluate this conclusion as an inductive argument. b. Evaluate this conclusion as a deductive argument.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Inductive versus Deductive Reasoning (cont.) Table 1 – Process Examples Positive IntegerArithmetic Result Solution a. We will begin by looking at some specific examples.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Inductive versus Deductive Reasoning (cont.) If we make the conclusion that choosing a positive integer, multiplying it by 2, and adding 1 to the product will result in an odd number simply by looking at these three examples, we are using inductive reasoning. We are making a generalization based on three specific examples. Although we know this to be valid, whenever we make a generalization, we are taking a risk that the generalized form will not always hold and therefore will be an invalid conclusion.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Inductive versus Deductive Reasoning (cont.) b.Now we will let the variable x represent the chosen integer and use deductive reasoning to show that we have indeed made a valid conclusion. Number: x Multiply by 2: 2x Add 1: 2x + 1 This means that, in algebraic form, the problem may be expressed as 2x + 1 (multiplying the chosen integer by 2 and adding 1).

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Inductive versus Deductive Reasoning (cont.) Using deductive reasoning, we know that when we multiply any number by 2, we will always get an even number. Thus, whenever we have an even number, as in 2x, and add 1, we will always get an odd number. Therefore, we can assert that, given any positive integer x, when x is multiplied by 2 and we add 1, the result will be an odd number. We can then use this general statement (which we have just proven to be true) and apply it to specific numbers.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #2 Consider the following process. Select a number and multiply the number by 10. Now subtract 25 from that product. Then divide by 5. Finally, subtract the original number from the result. Can you determine the general solution using deductive reasoning? Test the general solution with specific numbers.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #2 Answer: 2. To determine the general solution: Select a number: x; Multiply it by 10: 10x; Subtract 25 from the product: 10x  25; Divide by 5: 2x  5; Subtract the original value: 2x  5  x; This gives that the general result is x  5. Now we test the general solution with specific numbers. Select three numbers: 2, 10, 15; Multiply them by 10: 20, 100, 150; Subtract 25 from the product:  5, 75, 125; Divide by 5:  1, 15, 25; Subtract the original value:  3, 5, 10; Result is always 5 less than the original number.