Rational & Irrational Numbers Name7S1C3PO/7S1C3PO18S1C1PO2/8S1C1PO3/8S1C3PO2 Period

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Objective: 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 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?

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

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

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A. Real Numbers M08-S1C1-03

Organizers help us remember the real number system

Natural Numbers Whole Numbers Integers Rational Numbers Real NumbersThe Real Numbers can be split up into Rational and Irrational Numbers.

Tree Diagram

Venn Diagram

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Integer Subsets

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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

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power. a2 = a × aThe inverse of squaring is called "taking the square root."

Perfect Squares

12 1Example 22 432 = 3 × 3 = 9 32 9SO, the square root of 9 = 3 42 16(the symbol for square root looks like ) 52 25Rewrite the example above using the square root symbol 62 36

72 4982 64

The cube of a number, a, is that number taken to the third power. a3 = a × a × aThe inverse of cubing is called "taking the cube root".

92 81102 100112 121

Example 122 14423 = 2 × 2 × 2 = 8 132 169SO, the cube root of 8 = 2 142 196(the symbol for cube root looks like 3 ) 152 225Rewrite 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

1) What is the best estimate for 39?

A) 6.9282... Using the chart above, find the two perfect squares that are above and

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below 39. Use these squares to eliminate answer choices.

2 =

2 =

B) 5.9160...

C) 7.0710...

D) 6.2449…

E) 5.1961…

2) What is the best estimate for 5?

A 2.0037... B 1.7320...C 3.1209...D 2.2360...

3) What is the best estimate for 2?

A 1.7320... B 1.4142...C 3.1415...D 0.7320...

4) What is the best estimate for 27?

A 5.1961... B 5.0234... C 4.3498... D 3.9835...

5) What is the best estimate for 52?

A 10.7638... B 11.2343... C 8.2326... D 7.2111...

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

A) 2 2 =

5

2 = B) 3

C) 6

D) 10

E) 11

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

A 12

B 19

C 15

D 40

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

A 21

B 47

C 35

D 65

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

A 79

B 59

C 89

D 63

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

A 96

B 90

C 120

D 137

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.

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

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

A 7 and 8B 8 and 9C 9 and 10D 10 and 11

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

A 7 and 8B 8 and 9

C 9 and 10D 10 and 11

C. Classifying Real Numbers M08-S1C1-02

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N = natural W = whole I = integerR = rational C = irrational Y = real

N W I R C Y-2¾

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1) Which statement is true?

A 1 is not a whole number.B 2.5 is not an integer.C 4.8 is a whole number.D 3 is not a natural number.

2) Which statement is true?

A 1 is not a whole number.B 2.5 is not an integer.C 4.8 is a whole number.D 3 is not a natural number

3) Which statement is true?

A 1 is not a whole number.B 2.5 is not an integer.C 4.8 is a whole number.D 3 is not a natural number.

4) Which statement is true?

A 10 is a natural number.B -98 is a whole number.C -5 is a natural number.D 3.75 is an integer.

5) Which statement is true?

A -18 is an integer and a natural number.

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

6) Which statement is true?

A 14 is a natural number, a whole number, and an integer.B -25 is a natural number, a whole number, and an integer.C 3.6 is a natural number, a whole number, and an integer.D 0.78 is a natural number, a whole number, and an integer.

7) Which statement is true?

A 0 is a natural number, but not a whole number.B 67 is a natural number and a whole number.C 2.1 is a natural number, but not a whole number.D -8 is a natural number and a whole number.

8) Which of the following is an irrational number?

A QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

B QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

C QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

D QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

9) Which of the following is an irrational number?

A QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

B QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

C QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

D QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

11) Which set includes ONLY rational numbers?

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A QuickTime™ and aTIFF (Uncompressed) decompressor

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B QuickTime™ and aTIFF (Uncompressed) decompressor

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C QuickTime™ and aTIFF (Uncompressed) decompressor

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D QuickTime™ and aTIFF (Uncompressed) decompressor

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12) Which of the following groups listed below are subsets of the Irrational Numbers?

I. Real NumbersII. Natural Numbers

III. IntegersIV. Rational Numbers

V. Whole NumbersA III only

B None of the subsets listed are subsets of the Irrational Numbers.

C II, III, IV, and V only

D All of the subsets listed are subsets of the Irrational Numbers.

13) {2, 6, 18, 25, 30, 39, 50}The numbers shown above belong to which of the following subsets of the real numbers?

I. Irrational NumbersII. Integers

III. Whole NumbersIV. Natural NumbersV. Rational Numbers

A III and IV only

B II only

C II, III, and IV only

D II, III, IV, and V only

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