Mastering California s 15 Most Challenging Skills

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1 Mastering California s 15 Most Challenging Skills California Mathematics Golden Gate Bridge in San Francisco, California STUDENT NAME

2 Table of Contents Irrational Numbers (NS.1.4)... 4 Squares and Square Roots (NS.2.4)... 8 Estimating Square Roots (NS.2.4) Scientific Notation (NS.1.1) Adding and Subtracting Integers (NS.1.2) Multiplying and Dividing Integers (NS.1.2) Fractions (NS.1.2) Decimals (NS.1.2) Writing Expressions (AF.1.1) Writing Equations (AF.1.1) Solving Equations (AF.4.1) Distance, Rate, and Time (AF.4.2) Solving Inequalities (AF.4.1) Surface Area (MG.2.1) Volume (MG.2.1) Acknowledgments NS.1.4 CA At the beginning of each lesson, you will see a box with the shape of California and a content standard code in it. This code tells you what is being covered in the lesson. 3

$A rational number is any number that can be written as a fraction of two integers. The denominator of a rational number written as a fraction cannot be 0.$

3 NS.1.4 CA Irrational I Numbers Integers are whole numbers and their opposites. For example, 22, 21, 0, 1, and 2 are integers. A rational number is any number that can be written as a fraction of two integers. The denominator of a rational number written as a fraction cannot be 0. When the fraction is divided, it will be either a terminating decimal or a repeating decimal. Mammoth Lakes, California, near Mammoth Mountain, are a popular place any time of year for hiking, skiing, ice hockey and more. Examples of rational numbers are: 26, 0.5, 1 4, 20%, and An irrational number is a number that cannot be written as a fraction. Irrational numbers are non-terminating, non-repeating decimals. Pi (p) is an example of an irrational number. It is non-terminating and non-repeating. The square root of a number that is not a perfect square is also an irrational number. Example 1 4 Classify as a rational or irrational number. The bar notation over the decimal means those numbers repeat infinitely. It can be expressed as You can see a repeating pattern. This decimal is non-terminating, but is repeating. An irrational number is non-terminating and nonrepeating. Therefore, is a rational number. Build A Bridge 1 Classify each number as rational or irrational. 7 8 A: 2 B: C:

4 For calculations, p is given the approximation of 3.14 or 22. Although p is an 7 irrational number, 3.14 and 22 are rational numbers. This is because 3.14 is a 7 terminating decimal and 22 is a fraction of two integers. 7 Example 2 Which number is an irrational number? A 2 9 B C 16 D 5 Build A Bridge 2 Which number is an irrational number? A Choice A: 2 is a fraction of two integers. It is a 9 rational number. Choice B: is a repeating decimal. The ellipsis ( ) indicates the decimal does not end, and a pattern (86) is shown before the ellipsis. It is a rational number. B 47 C 100 D 3 16 Choice C: 16 is a perfect square. A perfect square is the square of an integer , so It is a rational number. Choice D: 5 is not a perfect square. There is no integer that when multiplied by itself equals 5. It creates a non-terminating and non-repeating decimal. It is an irrational number. 1 What is a rational number? Guided PractiCE 2 How is an irrational number different than a rational number? 5

5 3 Explain how some square roots can be rational numbers while others are irrational numbers is an approximation of p. Explain why the approximation of p is a rational 7 number. For Numbers 5 7, write the irrational number in each set , , 8, , 14, , , 6. 3, 49, p 16 8 Which number is a rational number? A 0. 6 B p C D 54 9 Which number is an irrational number? A B 20 C 144 D is a perfect square of which integer? A 32 B 14 C 9 D 8 6

6 Practice For Numbers 1 5, classify each number as rational or irrational Which number is a rational number? A p B C D Which number is a rational number? A B C 32 D Which number is an irrational number? A B C D p 9 Which number is an irrational number? A 81 B 25 C D The number 121 is a perfect square of which integer? A 9 B 10 C 11 D Gavin and his family traveled to Mammoth Lakes one summer. They played golf at the highest golf course in California. The highest hole is about 8,100 feet in elevation. 8,100 is a perfect square for which integer? A 40 B 81 C 90 D 405 7

or 5.00 or 5.000, and so on You can expand the decimal places of a number that already has digits to the right of the decimal point.

or 5.00 or 5.000, and so on You can expand the decimal places of a number that already has digits to the right of the decimal point. 1 LESSON Understanding Rational and Irrational Numbers UNDERSTAND All numbers can be written with a For example, you can rewrite 22 and 5 with decimal points without changing their values. 22 5 22.0 or