Copyright © Cengage Learning. All rights reserved. CHAPTER 1 Foundations for Learning Mathematics.

Copyright © Cengage Learning. All rights reserved.

CHAPTER 1

Foundations for Learning Mathematics

Copyright © Cengage Learning. All rights reserved.

SECTION 1.3

Reasoning and Proof

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What Do You Think?

• Do you think young children can do proofs?

• How can the classroom nurture intuitive reasoning?

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Reasoning and Proof

This was one of the hardest sections to write, partly because there are so many different kinds of reasoning and partly because there are so many different ways that it is taught and developed.

We will investigate three types of reasoning: inductive reasoning, deductive reasoning, and intuitive reasoning. We will also discuss analytic reasoning.

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Investigation A – Does Your Answer Make Sense?

All 261 fifth-graders in a school are going on a field trip, and each bus can carry 36 children and 4 adults. How many buses are needed?

Discussion:If you divide 261 by 36, you get 7.25. When this problem was given to seventh-graders in one of the National Assessments of Educational Progress, a majority of children gave 7 as the answer.

Why do think they did? Many of them had been taught the rules of rounding mechanically, and thus they rounded 7.25 to 7.

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Investigation A – Discussion

While the mathematical rule of rounding is to round 7.25 down to 7, the real-life application of this computation requires us to round 7.25 up to 8.

When solving problems, it is always necessary not only to check your answer but your reasoning: Does the answer make sense? Does the reasoning make sense?

cont’d

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Inductive Reasoning

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Inductive Reasoning

Inductive reasoning is a process of coming to a general conclusion from seeing patterns in specific examples and looking for the regularity in those patterns.

This kind of reasoning is crucial in the child’s construction of the world, and it involves making generalizations from seeing patterns in specific examples.

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Investigation B – Inductive Thinking with Fractions

Let’s say you don’t know how to add fractions with different denominators. Look at the following three examples. Can you see a pattern that would enable you to add other fractions?

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Investigation B – Discussion

We can describe the pattern in words by saying that whenever you have two fractions with a 1 in the numerator, the numerator of the sum is found by adding the two denominators, and the denominator of the sum is determined by multiplying the two denominators.

More formally, we would say that when you add two unit fractions whose denominators are relatively prime, the numerator of the sum is equal to the sum of the denominators of the two fractions, and the denominator of the sum is equal to the product of the denominators of the two fractions.

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Investigation B – Discussion

Unit fractions are fractions whose numerators are 1, and two numbers that are relatively prime have no common factors other than 1.

The pattern also works when the denominators are not relatively prime, but then the answer is not in simplest form; an example is

The result can be expressed more succinctly with notation and proves this conjecture for all cases, not just some.

cont’d

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Investigation B – Discussion

Look at Figure 1.5 below.

If we draw two points on a circle and connect them, we create 2 regions.

If we draw three points on a circle and connect each point to every other point, we create 4 regions. If we continue this with four points, we create 8 regions. If we continue this with five points, we create 16 regions.

cont’d

Figure 1.5

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Investigation B – Discussion

If we draw six points and connect each point to every other point, how many regions will be created?

Most people conclude that the number of regions doubles each time. Alas, this is not so. This pattern breaks down with six points.

Try as you might (and believe me, mathematicians have tried), no matter where those six points are placed, the maximum number of regions formed is 31.

This is a classic illustration of the need to be careful with patterns and with inductive reasoning: Patterns do not always hold.

cont’d

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Deductive Reasoning

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Deductive Reasoning

Deductive reasoning is a process of reaching a conclusion from one or more statements, called the hypotheses.

An argument is a set of statements in which the last statement is called the conclusion and there is one or more hypotheses.

Statements of the form “if p, then q” are called conditional statements. The “if” part of a conditional is called the hypothesis of the implication, and the “then” part is called the conclusion.

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Deductive Reasoning

A statement and its converse are often not both true. Consider some examples.

There are two other statements that have an interesting relationship to the original statement. Let us consider the statement here to be the true one: If it is a dog, then it has four legs and a tail. We can write the converse, inverse, and contrapositive of any if–then statement.

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Deductive Reasoning

The second column of Table 1.12 shows the converse, inverse, and contrapositive in everyday English; the third column shows the statements in shorthand, where the hypothesis is denoted by p and the conclusion by q; and the fourth column shows the statements in their most succinct form.

Table 1.12

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Deductive Reasoning

It turns out that a statement and its contrapositive are logically equivalent. That is, if a statement is true, the contrapositive of that statement is also true.

There are laws of deductive reasoning that enable us to determine whether the reasoning in an argument is valid or invalid.

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Deductive Reasoning

Below are the four possibilities, stated in their general form at the left and with a mathematical example at the right.

Case 1 and Case 2 show examples of valid arguments, while Case 3 and Case 4 illustrate two common invalid arguments.

Case 1: In this case, we have a general statement that has been determined to be true, and we have a specific instance (example) of the first part of the statement. In this case, the conclusion is true.

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Deductive Reasoning

This law of logic is called the Law of Detachment. It is also called affirming the hypothesis.

If p, then q. If a figure is a square, then it has four sides.

p is true. This figure is a square.

Then q is true. Therefore, this figure must have four sides.

Case 2: This law of logic is called Modus Tollens. It is also called denying the conclusion.

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Deductive Reasoning

Essentially, this is a restatement of what we saw earlier:

The contrapositive of a true statement is also true. That is, if we know that p q then we know that ~q ~p.

If p, then q. If a figure is a square, then it has four sides.

q is not true. This figure does not have four sides.

Then p is not true. Therefore, this figure is not a square.

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Deductive Reasoning

Case 3 and Case 4 show examples of two common mistakes (equating a statement and its converse or inverse) that lead to invalid arguments. In each of the cases, we can demonstrate the falsity of the conclusion with a counterexample.

Case 3: This kind of error in reasoning is called the fallacy of affirming the consequence.

If p, then q. If a figure is a square, then it has four sides.

q is true. This figure has four sides.

Then p is true. Therefore, this figure is a square.

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Deductive Reasoning

One counterexample is a trapezoid, which has four sides but is not necessarily a square.

Case 4: This kind of error in reasoning is called the fallacy of denying the antecedent.

If p, then q. If a figure is a square, then it has four sides.

p is not true. This figure is not a square.

Then q is not true. Therefore, this figure does not have four sides.

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Deductive Reasoning

One counterexample is a parallelogram, which has four sides but is not necessarily a square.

An important kind of reasoning that is necessary for success in mathematics as well as many other fields is called analytic thinking.

A common definition of analytic thinking is to break acomplex idea into its component parts and theirrelationships.

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Deductive Reasoning

Venn diagrams:

We can also represent conditional statements with Venn diagrams.

The Venn diagram in Figure 1.6 represents the valid relationship between all animals, those with 4 legs and a tail, and dogs.

Figure 1.6

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Deductive Reasoning

All animals that have four legs and a tail that are not dogs are inside the large circle but outside the dog circle.

All other animals are inside the square but outside the circles.

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Deductive Reasoning

The four Venn diagrams in Figure 1.7 illustrate the four statements in Table 1.12.

Statement Converse Inverse Contrapositive

Figure 1.7

If p, then q.p → q

If it is a dog, it has four legs.

If q, then p. q → p If it has four legs, and a tail, it is a dog.

If not p, then not q. ~p → ~qIf it is not a dog, it doesn't have four legs and a tail.

If not q, then not p. ~q → ~pIf it doesn't have four legs and a tail,it is not a dog.

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Deductive Reasoning

The first Venn diagram represents the true statement. Because all dogs have four legs and a tail, the small circle is entirely within the larger circle.

The second Venn diagram illustrates why the converse is not necessarily true.

There are two regions (denoted by the x’s) that satisfy the statement “animals having four legs and a tail”: inside the smaller circle and inside the larger circle but outside the smaller circle.

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Deductive Reasoning

This latter region is where the counterexamples come from: sheep, cows, horses, etc.

The third Venn diagram illustrates why the inverse is not necessarily true.

There are two regions (denoted by the x’s) that satisfy the statement “animals that are not dogs”: inside the larger circle but outside the smaller circle and outside both circles.

This former region is where the counterexamples come from. That is, sheep, cows, and horses are not dogs, but they do have four legs and a tail.

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Deductive Reasoning

The fourth Venn diagram illustrates why the contrapositive is true.

There is only one region for “doesn’t have four legs and a tail”: outside both circles. And all animals in this region (such as chickens and fish) are definitely not dogs.

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Investigation C – Deductive Reasoning and Venn Diagrams

Let us investigate two arguments to see how Venn diagrams can help determine the validity of an argument.

For each argument, first represent the situation with a Venn diagram and determine whether the argument is valid or not valid. Then, determine whether the argument matches case 1, 2, 3, or 4.

Argument 1:

If a quadrilateral is a square, then the diagonals are perpendicular.

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Investigation C – Deductive Reasoning and Venn Diagrams

The diagonals of this quadrilateral are perpendicular.

Therefore this quadrilateral is a square.

Argument 2:

If a number is divisible by 4, then it is also divisible by 2.

79 is not divisible by 2.

Therefore, 79 is not divisible by 4.

cont’d

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Investigation C – Discussion

The first argument fits Case 3, and thus the conclusion is not valid.

If p, then q.

q is true.

Therefore, p is true.

It can be represented by the Venn diagram in Figure 1.8.

Figure 1.8

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Investigation C – Discussion

The second argument fits Case 2, and thus the conclusion is valid.

If p, then q.

q is not true.

Therefore, p is not true.

It can be represented by theVenn diagram in Figure 1.9.

Figure 1.9

cont’d

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Deductive Reasoning

Biconditional statements:

There is one other concept from deductive reasoning that we will discuss in this chapter: the biconditional statement.

A biconditional statement occurs when a statement and its converse are both true.

This is especially relevant in mathematics: Most mathematical definitions are biconditional statements.

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Deductive Reasoning

Consider the following two statements:

If two lines intersect to form right angles, then they are perpendicular.

If two lines are perpendicular, then they intersect to form right angles.

In this case, the statement and its converse are always both true. We can use “if and only if” language to combine them into the following biconditional statement, which is also the definition of perpendicular:

Two lines are perpendicular iff they intersect to form right angles.

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Investigation D – Why Is the Sum of Two Even Numbers an Even Number?

Consider the question of how to prove that the sum of two even numbers is even.

Examine the following responses by the students and think if they constitute a proof:

Paul: I know that the sum is even because my older sister told me it always happens that way.

Zoe: I know it will add to an even number because4 + 4 = 8 and 8 + 8 = 16.

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Investigation D – Why Is the Sum of Two Even Numbers an Even Number?

Evan: We really can’t know! Because we might not know about an even number and if we add it with 2 it might equal an odd number!

Melody: (Pointing to two sets of cubes she had arranged) This number is in pairs (pointing to the light-colored cubes), and this number is in pairs (pointing to the dark-colored cubes), and when you put them together, it’s still in pairs.

cont’d

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Investigation D – Discussion

Shifter describes four categories of justification common to elementary students (and I find with college students too):

appeal to authority,

inference from instances,

assertion that claims about an infinite class cannot be proven, and

reasoning from representation or context.

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Investigation D – Discussion

Do you see these categories in the students above? In the diagrams below, you can see how Melody’s assertion closely parallels a more formal proof.

An integer is even if it can A number is even if it can represented as 2 times be broken up into 2 pairs.another integer.

If a and b are even These two numbers, are evennumbers, then we because they can be broken can find two integers x into pairs.and y such that a = 2x and b = 2y.

cont’d

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Investigation D – Discussion

So a + b = 2x + 2y; If you put them together, but then a + b = 2(x + y) you still have 2 pairs.

Thus a + b is equal to 2 times Therefore the sum of the an integer, but that is the two number is also even. definition of an even number.

cont’d

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Investigation D – Discussion

Schifter asserts that young children are capable of making and justifying mathematical generalizations and that making arguments from representations (physical objects, pictures, diagrams, or story contexts) is an effective way to help students develop such reasoning capacity.

She proposed three criteria for such representations:

1. The meaning of the operation(s) involved is represented in diagrams, manipulatives, or story contexts.

2. The representation can accommodate a class of examples.

cont’d

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Investigation D – Discussion

3. The conclusion of the claim follows from the structure of the representation.

Do you see how Melody’s argument satisfied these criteria?

1. Her representation modeled two whole numbers.

2. Her language did not say 10 + 16 but rather two whole numbers. That is, her argument did not depend on the actual value of the two numbers (as Zoe’s did).

3. When you place the two diagrams together, the resulting amount can also be represented in pairs.

cont’d

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Investigation E – Darts, Proof, and Communication

Suppose you have a dart board like the one in Figure 1.10. You throw four darts, all of which land on the dart board.

What kinds of scores would be possible and what kinds of scores would be impossible?

Figure 1.10

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Investigation E – Discussion

After a few minutes, one of the students, Erika, suddenly said, “Only even numbers are possible.” and her reasoning is, “Well I know that an odd plus an odd is even and an odd plus an even is odd. [At this point, she held up four fingers to represent the four darts.]

The first two darts are odd and so when you add them, you have an even number. [She joined two of her fingers together to indicate the combined score from two darts.]

Now this number (even) plus the next dart (odd) will make an odd number. [She now joined three of her fingers together to indicate the combined score from the first three darts.]

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Investigation E – Discussion

Now this number (odd) plus the last dart (odd) will make an even number. So the only possible scores you can get are even numbers.”

We can represent Erika’s proof as shown in Figure 1.11.

cont’d

Figure 1.11

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Intuitive Reasoning

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Intuitive Reasoning

There is one other type of reasoning that occurs a lot in mathematics.

Intuitive reasoning is not well understood, and we know very little about how to “teach” it. Let’s look at a couple of examples.

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Investigation F – The Nine Dots Problem

Not all problem solving involves computation and formulas, as this investigation shows.

Without lifting your pencil, can you go through all nine dots in Figure 1.12 with only four lines?

Figure 1.12

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Investigation F – Discussion

This is a very famous problem, which some of you may have already encountered because of its moral: This problem is impossible to solve as long as you “stay inside the box.” In order to solve the problem, you need to go “outside the box.”

This idea of not getting stuck inside the box is crucial to good problem solving.

In many real-life problems, the solution to a problem requires that people think about the problem differently.

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Investigation F – Discussion

In this case, inductive reasoning is not very relevant. We don’t have other examples before us from which we can generalize.

Classic deductive reasoning is not applicable here. At some point, either out of desperation or after concluding that no solution within the square is possible, some people solve this puzzle by trying solutions that go outside the boundaries of the invisible square surrounding this set of dots.

This is similar to figuring out the nth term of the 4, 9, 19, 39, 79, . . . sequence by “feeling” its connectedness to the 5, 10, 20, 40, 80, . . . sequence.

cont’d

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Investigation G – How Many Games in the Tournament?

In the NCAA women’s basketball tournament, 64 teams are selected in a single elimination tournament. How many games are in the tournament?

Discussion:The following solution nicely illustrates inductive reasoning. I had given this question to a friend who cocked her head for a moment and then said, “63 games.”

I was flabbergasted and asked, “How did you solve it so quickly and without any paper?” Her reply: “If one team is the champion, then 63 teams have to lose.”

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Investigation G – Discussion

As in the previous problem, she had “gotten out of the box” and had seen the problem from a very different perspective.

A more traditional solution path, one that I had taken, looks like this:

Round Games

1 32

2 16

3 8

4 4

5 2

6 1

cont’d