1-2 Irrational Numbers and Square Roots

Geogebra Finding square roots of perfect squares Estimating a square root Comparing square roots Surface area of a sphere Classifying real numbers

Simplifying square roots Classifying real numbers as rational or irrational Video Tutor Help

1-2 Videos Use squares to find side length of a square In this lesson, you will learn how to find the side length of a square given its area by using what you already know about squares to guess and check. Standards: 8.EE.A.2 Use visuals to understand and generate perfect squares In this lesson, you will learn how to generate perfect squares by exploring a visual and learning a definition. Standards: 8.EE.A.2 Find the square root of a perfect square In this lesson, you will learn how to find the square root of perfect squares by visualizing squares and learning new notation. Standards: 8.EE.A.2 Understand the relationship between squaring and taking the square root In this lesson you will learn about the relationship between squaring and taking the square root by exploring a visual and definitions you already know. Standards: 8.EE.A.2

Solve equations with squares and square roots In this lesson you will learn how to solve equations involving squares and square roots by using properties you already know. Standards: 8.EE.A.2 Identify perfect squares and find square roots In this lesson you will learn how to identify perfect squares and find square roots. Standards: 8.EE.A.2 Find square roots of perfect squares In this lesson you will learn how to find square roots of perfect squares. Standards: 8.EE.A.2 Simplify expressions involving square roots In this lesson you will learn how to simplify expressions involving square roots. Standards: 8.EE.A.2 Solve problems with inverse of squares and square roots In this lesson you will learn how to solve problems with inverse of squares and square roots. Standards: 8.EE.A.2

Find the side lengths of a square given its area In this lesson you will learn how to find the side lengths of a square given its area. Standards: 8.EE.A.2 Understand and apply the definition of irrational numbers In this lesson you will learn how to understand and apply the definition of irrational numbers. Standards: 8.NS.A.1 Distinguish between rational and irrational numbers In this lesson you will learn how to distinguish between rational and irrational numbers. Standards: 8.NS.A.1 Compare irrational and rational numbers In this lesson you will learn how to compare irrational and rational numbers. Standards: 8.NS.A.1

Place nonperfect square roots between 2 integers In this lesson you will learn how to place nonperfect square roots between 2 integers. Standards: 8.NS.A.2

Video Tutor Help Irrational Numbers Classifying Real Numbers Khan Academy Word Problem: given area find side length Word problem: square roots Estimating square roots Classifying real numbers as rational or irrationalClassifying real numbers as rational or irrational (1-1 and 1-2) Simplifying square rootsSimplifying square roots (1-2) Irrational Numbers and Square Roots Course 3 Rational Exponents Exponents can not only be integers and unit fractions. An exponent can be any rational number expressed as the quotient of two integers. Irrational Numbers and Square Roots Course 3 Simplifying Radical Expressions Radical expressions can often be simplified by moving factors which are perfect roots out from under the radical sign. 1-2 Irrational Numbers and Square Roots Course 3 Irrational Numbers Although the Greeks initially thought all numeric qualities could be represented by the ratio of two integers, i.e. rational numbers, we now know that not all numbers are rational. How do we know this? Irrational Numbers and Square Roots Course 3 Real Numbers There are an infinite number of rational numbers, but there are infinitely more irrational numbers. Neither type of number can represent every type of numeric quantity. By combining the rational and irrational numbers into the real numbers, a continuum of numbers is created which can represent any quantity in the real WhyU: Roots and Unit Fraction Expenents

Use squares to find side length of a square In this lesson, you will learn how to find the side length of a square given its area by using what you already know about squares to guess and check. Standards: 8.EE.A.2 Use visuals to understand and generate perfect squares In this lesson, you will learn how to generate perfect squares by exploring a visual and learning a definition. Standards: 8.EE.A.2 Find the square root of a perfect square In this lesson, you will learn how to find the square root of perfect squares by visualizing squares and learning new notation. Standards: 8.EE.A.2 Understand the relationship between squaring and taking the square root In this lesson you will learn about the relationship between squaring and taking the square root by exploring a visual and definitions you already know. Standards: 8.EE.A.2 Solve equations with squares and square roots In this lesson you will learn how to solve equations involving squares and square roots by using properties you already know. Standards: 8.EE.A.2

Identify perfect cubes and find cube roots In this lesson you will learn how to identify perfect cubes and find cube roots. Standards: 8.EE.A.2 Solve equations with cubes and cube roots In this lesson you will learn how to solve equations with cubes and cube roots. Standards: 8.EE.A.2

Identify perfect squares and find square roots In this lesson you will learn how to identify perfect squares and find square roots. Standards: 8.EE.A.2 Find square roots of perfect squares In this lesson you will learn how to find square roots of perfect squares. Standards: 8.EE.A.2 Simplify expressions involving square roots In this lesson you will learn how to simplify expressions involving square roots. Standards: 8.EE.A.2 Solve problems with inverse of squares and square roots In this lesson you will learn how to solve problems with inverse of squares and square roots. Standards: 8.EE.A.2 Solve equations with squares and square roots In this lesson you will learn how to solve equations with squares and square roots. Standards: 8.EE.A.2 Find the side lengths of a square given its area In this lesson you will learn how to find the side lengths of a square given its area. Standards: 8.EE.A.2

Understand and apply the definition of rational numbers In this lesson you will learn how to understand and apply the definition of rational numbers. Standards: 8.NS.A.1 Understand and apply the definition of irrational numbers In this lesson you will learn how to understand and apply the definition of irrational numbers. Standards: 8.NS.A.1 Distinguish between rational and irrational numbers In this lesson you will learn how to distinguish between rational and irrational numbers. Standards: 8.NS.A.1

Understand and apply the definition of rational numbers In this lesson you will learn how to understand and apply the definition of rational numbers. Standards: 8.NS.A.1 Understand and apply the definition of irrational numbers In this lesson you will learn how to understand and apply the definition of irrational numbers. Standards: 8.NS.A.1 Distinguish between rational and irrational numbers In this lesson you will learn how to distinguish between rational and irrational numbers. Standards: 8.NS.A.1 Compare irrational and rational numbers In this lesson you will learn how to compare irrational and rational numbers. Standards: 8.NS.A.1

Convert repeating decimals into fractions In this lesson you will learn how to convert repeating decimals into fractions. Standards: 8.NS.A.1 Place nonperfect square roots between 2 integers In this lesson you will learn how to place nonperfect square roots between 2 integers. Standards: 8.NS.A.2

Worksheets 1-2 Note-Taking Guide 1-2 Practice 1-2 Guided Problem Solving

Vocabulary Practice Chapter 1 Vocabulary (Electronic) Flash Cards

Additional Lesson Examples 1-2 Step-by-Step Examples

Lesson Readiness 1-2 Problem of the Day 1-2 Lesson Quiz

Equivalent Forms of Rational Numbers Write each as a fraction in simplest form – 3. Write as a decimal. Write each decimal as a mixed number or fraction in simplest form LESSON – Lesson Quiz

Finding square roots of perfect squares

4  4 = 4 2 = 16 = 4= 4 Positive square root of 16 (–4)(–4) = (–4) 2 = 16 = –4 Negative square root of 16 – A number that is multiplied by itself to form a product is a square root of that product. The radical symbol is used to represent square roots. For nonnegative numbers, the operations of squaring and finding a square root are inverse operations. In other words, for x ≥ 0, Positive real numbers have two square roots.

A perfect square is a number whose positive square root is a whole number. Some examples of perfect squares are shown in the table The principal square root of a number is the positive square root and is represented by. A negative square root is represented by –. The symbol is used to represent both square roots.

The small number to the left of the root is the index. In a square root, the index is understood to be 2. In other words, is the same as. Writing Math

Find the two square roots of 81. Exploring Square Roots and Irrational Numbers LESSON = 81 –9 (–9) = 81 The two square roots of 81 are 9 and –9. Additional Examples

Example 1-1a Find indicates the positive square root of 81. Since Answer: Find a Square Root

Example 1-1b Answer: 8 Find

Example 1-2a Find indicates the negative square root of 144. Answer: Since Find the Negative Square Root

Example 1-2b Find Answer: ─ 6

Estimate the value of – 70 to the nearest integer. Exploring Square Roots and Irrational Numbers LESSON 1-2 Since 70 is closer to 64 than it is to 81, – 70 –8. Additional Examples

Estimating a square root

Math fact: The sum of any number of consecutive odd whole numbers, beginning with 1, is a perfect square e.g. 1+3=4, 1+3+5=9, =16

Example 1: Estimating Square Roots of Numbers The 55 is between two integers. Name the integers. Explain your answer. List perfect squares near 55.36, 49, 64, < 55 < 64 Find the square roots of the perfect squares is between 7 and 8 because 55 is between 49 and 64. Find the perfect squares nearest < 55 < 64 7 < 55 < 8

Example 2: Approximating Square Roots to the Nearest Hundredth 121 < 135 < 144 The whole number part of the answer is 11. Find the perfect squares nearest 135. Find the square roots of the perfect squares. √121 < √135 < 144 √ 11 < 135< 12 √ Step 1 Find the value of the whole number. The number will be between 11 and 12. Approximate √135 to the nearest hundredth.

Example 2 Continued 135 – 121 = ÷ 23 ≈ Find the difference between the given number, 135, and the lower perfect square. Approximate √135 to the nearest hundredth. Step 2 Find the value of the decimal. Write the difference as a ratio. 144 – 121 = 23 Find the difference between the greater perfect square and the lower perfect square Divide to find the approximate decimal value.

Example 2 Continued = Combine the whole number and decimal. Step 3 Find the approximate value ≈ Round to the nearest hundredth. The approximate value of 135 to the nearest hundredth is Approximate √135 to the nearest hundredth.

Example 3: Using a Calculator to Estimate the Value of a Square Root Use a calculator to find 600. Round to the nearest tenth. 600 ≈ ≈ … Use a calculator. Round to the nearest tenth. 600 rounded to the nearest tenth is 24.5.

The math class drops a small ball from the top of a stairwell. They measure the distance to the basement as 48 feet. Use the formula d = 16t 2 to find how long it takes the ball to fall. Exploring Square Roots and Irrational Numbers LESSON = 16t 2 Substitute 48 for d. d = 16t 2 Use the formula. = t 2 Divide each side by = t 2 Simplify. Additional Examples

(continued) Exploring Square Roots and Irrational Numbers LESSON 1-2 It takes about 1.7 seconds for the ball to fall 48 ft. 3 = tFind the positive square root. Use a calculator Round to the nearest tenth. 1.7 t Additional Examples

Example 1-3a ALGEBRA Solve Write the equation. Take the square root of each side. The equation has two solutions, Answer: Notice that Use Square Roots to Solve an Equation

Example 1-3b Answer: ALGEBRA Solve

Identify each number as rational or irrational. Explain. Exploring Square Roots and Irrational Numbers LESSON 1-2 d Irrational; the decimal does not terminate or repeat. Rational; the decimal repeats.a. – b Rational; the number can be written as the ratio Irrational; 90 is not a perfect square. c. 90 Additional Examples

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