What you should learn

To find square roots

To classify numbers

To graph solutions of inequalities on number lines

NCTM Curriculm Standards 1, 2, 4, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Square Roots

Perfect Square

Radical Sign

Principal Square Root

Irrational Numbers

Real Numbers

Completeness Property

Introduction: Sweden botanist Carolus Linnaeus (1707 - 1778) developed the system we use today to classify every kind of living thing according to common characteristics. For example, an African elephant is form the Mammalia Class and also from the Animalia Kingdom.

In mathematics, we classify numbers that have common characteristics. So far in this text, we have classified numbers as natural numbers, whole numbers, integers, and rational numbers.

The square roots of perfect squares are classified as rational numbers. A square root is one of two equal factors of a number. For example, one square root of 81 is 9 since 9 * 9 or 9 is 81. A rational number, like 81, whose square root is a rational number, is called a perfect square.

It is also true that -9 * (-9) = 81. Therefore, -9 is another square root of 81.

9 = 9 * 9 = 81

(-9) = (-9)(-9) = 81

Definition of Square Root: If x= y, then x is a square root of y.

 

The symbol , called a radical sign, is used to indicate a nonnegative or principal square root of the expressions under the radical sign.

81 = 9 where indicates the principal square root of 81

-81 = -9 where -81 indicates the negative square root of 81

81 = 9 where 81 indicates both square roots of 81

Exercise 1: Find each square root.

a. 25

The symbol25 represents the principal square root of 25.

Since 5 = 25, you know that 25 = 5

b. -144

c. 0.16

 

 

Most scientific calculators have a square root key. When you press this key, the number in the display is replaced by its principal square root.

Exercise 2: Use a scientific calculator to evaluate each expression if x = 2401, a = 147, and b = 78.

a. x

b. (a + b)

 

 

A number such as 2 is the square root of a number that is not a perfect square. Notice what happens when you find 2 with a calculator (1.412136...)

The decimal form of 2 appears to continue indefinitely wihtout any pattern of repreating digits. So, it seems that 2 is not a rational number. We do not assume that 2 is not rational; rather, to be sure, it must be proven. Numbers that cannot be expressed as repeating or terminating decimals are called irrational numbers.

Definition of an Irrational Number: An irrational number is a number that cannot be expressed in the form a/b, where a and b are integers and b0

The set of rational numbers and the set of irrational numbers together form the set of real numbers. The Venn diagram below shows the relationships among natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

Exercise 3: Name the set or sets of numbers to which each real number belongs.

a. 0.833333333...

This repeating decimal is a rational number since it is equivalent to 5/6.

b. -16

c. 14/2

d 120

 

The solutions tomany real-world problems are irrational numbers.

Exercise 4: The area of a square is 325 square inches. Find its perimeter to the nearest hundredth.

You have graphed rational numbers on number lines. Yet, if you graphed all of hte rational numbers, the number line would still not be complete. The irrational numbers complete the number line. The graph of all real numbers is the entire number line. This is illustrated by the completeness property.

Completeness Property for Points on the Number Line: Each real number corresponds to exactly one point on the number line. Each point on the number line corresponds to exactly one real number.

 

Recall that equations like x - 5 = 11 are open sentences. Inequalities like x < 6 are also considered to be open sentences. To solve x < 6, determine what replacemetns for x make x < 6 true. All numbers less than 6 make the inequality true. This can be shown by the solution set {real numbers less than 6}. Not only does this include integers like 3, 0, and -4, but it also includes all rational numbers less than 6 such as 1/2, -5 3/8, and -3 and all irrational numbers less than 6 such as 5, 3, and .

Exercise 5: Graph each solution set

a. y -7

The heavy arrow indicates that all numbers to the right of -7 are included. The dot indicates that the point corresponding to -7 is included in the graph of the solution set.

b. p3/4

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 21 - 65 odd, 67 - 78

Alternative Homework: Enriched: 22 - 60 even, 61 - 78

Extra Practice: Students book page 761 Lesson 2-8

Extra Practice Worksheet: Click Here.

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