8 S1c1po3 Rational & Irrational

Rational & Irrational Numbers 7S1C3PO/7S1C3PO1 8S1C1PO2/8S1C1PO3/8S1C3PO2

Name Period

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Date

Objective:

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Students are able to justify the location of real numbers (rational and irrational) on a number line by estimating using models (number lines), symbols, and written words

ESSENTIAL QUESTIONS

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What is the difference between a perfect square and a square root?



How can you use estimation strategies, such as benchmark numbers, to estimate and calculate a solution for a problem? How do you know your estimate is reasonable? How can you justify your solution?



What is the difference between rational and irrational numbers?

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How can the real number system be described using a graphic organizer?

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How can rational numbers be used to help locate irrational numbers? Why is the location of an irrational number an approximate location?

Vocabulary: Real Numbers All the numbers, if you can see it, it is real

Rational Numbers Fractions, any number you can make a fraction, decimals that terminate or have a pattern Integers Negative numbers, zero, and positive numbers

Whole Zero, and positive numbers

Natural Positive numbers

Irrational

Never stop, have no pattern, crazy decimals 1

A. Real Numbers M08-S1C1-03

Organizers help us remember the real number system Natural Numbers Whole Numbers Integers Rational Numbers The Real Numbers can be split up into Rational and Irrational Numbers.

Real Numbers

Tree Diagram

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Venn Diagram

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B. Estimating Square Roots

M07-S1C3-03 & M08-S1C3-02

The square of a number, a, is that number taken to the second power. a2 = a × a The inverse of squaring is called "taking the square root." Example 32 = 3 × 3 = 9 SO, the square root of 9 = 3 (the symbol for square root looks like √ ) Rewrite the example above using the square root symbol

The cube of a number, a, is that number taken to the third power. a3 = a × a × a The inverse of cubing is called "taking the cube root". Example 23 = 2 × 2 × 2 = 8 SO, the cube root of 8 = 2 (the symbol for cube root looks like 3√ )

Perfect Squares

12 22 32 42 52 62 72 82 92 102 112 122 132 142 152

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225

Rewrite the example above using the cube root symbol

We use the perfect squares to estimate the location of some numbers A) Identify on the number line below where the √4 and √9 are located B) Estimate where the √8 would be located

C) Estimate where the √5 would be located

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*The square root of numbers that are NOT perfect squares are IRRATIONAL 4

1) What is the best estimate for A)

6.9282...

B)

5.9160...

C)

7.0710...

D)

6.2449…

E)

5.1961…

2)

3)

=

2

=

2?

1.7320... 1.4142... 3.1415... 0.7320... 27?

5.1961... 5.0234... 4.3498... 3.9835...

What is the best estimate for A B C D

5?

2.0037... 1.7320... 3.1209... 2.2360...

What is the best estimate for A B C D

5)

2

What is the best estimate for A B C D

4)

Using the chart above, find the two perfect squares that are above and below 39. Use these squares to eliminate answer choices.

What is the best estimate for A B C D

39?

52?

10.7638... 11.2343... 8.2326... 7.2111... 5

1) Which of the following numbers is between 2 and 3? A)

√2

B)

√3

C)

√6

D)

√10

E)

√11 2)

3)

4)

5)

2

=

2

=

The square root of which of the following numbers is between 4 and 5? A

12

B

19

C

15

D

40

The square root of which of the following numbers is between 6 and 7? A

21

B

47

C

35

D

65

The square root of which of the following numbers is between 8 and 9? A

79

B

59

C

89

D

63

The square root of which of the following numbers is between 10 and 11? A

96

B

90

C

120

D

137 6

1) A part of the real number line is shown below.

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2) A part of the real number line is shown below. QuickTimeª and a TIFF (Uncompressed) decompressor are needed to see this picture.

3) A part of the real number line is shown below. QuickTimeª and a TIFF (Uncompressed) decompressor are needed to see this picture.

4)

The square root of 79 is between which two whole numbers?

A B C D 5)

7 and 8 8 and 9 9 and 10 10 and 11

The square root of 112 is between which two whole numbers?

A B C D

7 and 8 8 and 9 9 and 10 10 and 11

C. Classifying Real Numbers

M08-S1C1-02

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N = natural R = rational N

W = whole C = irrational W

I

I = integer Y = real R

C

Y

-2 ¾ √25 √5 4 0 π 1) Which statement is true? A B C D

1 is not a whole number. 2.5 is not an integer. 4.8 is a whole number. 3 is not a natural number.

2) Which statement is true? A B C D

1 is not a whole number. 2.5 is not an integer. 4.8 is a whole number. 3 is not a natural number

3) Which statement is true? A B C D

1 is not a whole number. 2.5 is not an integer. 4.8 is a whole number. 3 is not a natural number.

4) Which statement is true? A B C D

10 is a natural number. -98 is a whole number. -5 is a natural number. 3.75 is an integer.

5) Which statement is true? A

-18 is an integer and a natural number. 8

B C D

5.7 is an integer and a natural number. -2.3 is an integer and a natural number. 9 is an integer and a natural number.

6) Which statement is true? A B C D

14 is a natural number, a whole number, and an integer. -25 is a natural number, a whole number, and an integer. 3.6 is a natural number, a whole number, and an integer. 0.78 is a natural number, a whole number, and an integer.

7) Which statement is true? A B C D

0 is a natural number, but not a whole number. 67 is a natural number and a whole number. 2.1 is a natural number, but not a whole number. -8 is a natural number and a whole number.

8) Which of the following is an irrational number?

A

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B

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C

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D

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9) Which of the following is an irrational number?

A

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B

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C

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D

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11) Which set includes ONLY rational numbers? 9

A B C D

12)

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Which of the following groups listed below are subsets of the Irrational Numbers? I. II. III. IV.

A B C D 13)

A B C D

Real Numbers Natural Numbers Integers Rational Numbers

V. Whole Numbers III only None of the subsets listed are subsets of the Irrational Numbers. II, III, IV, and V only All of the subsets listed are subsets of the Irrational Numbers. {2, 6, 18, 25, 30, 39, 50} The numbers shown above belong to which of the following subsets of the real numbers? I. Irrational Numbers II. Integers III. Whole Numbers IV. Natural Numbers V. Rational Numbers III and IV only II only II, III, and IV only II, III, IV, and V only

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