8/27/15

1.1. Please complete the Please complete the “conclusion” questions on the “conclusion” questions on the back of your scavenger hunt.back of your scavenger hunt.

2.2. Share with a neighbor.Share with a neighbor.

3.3. Let’s share out.Let’s share out.

Making Sense of Rational and Irrational Numbers

Essential Question: How are Essential Question: How are rational and irrational rational and irrational numbers simplified?numbers simplified?

Biologists classify animals based on shared characteristics. The horned

lizard is an animal, a reptile, a lizard, and a gecko!

Numbers can also be classified!

AnimalReptile

LizardGecko

The set of real numbers is all numbers that can be written on a number line.

It consists of 2 subsets – rational numbers and irrational numbers.

Irrational numbersRational numbers

Real Numbers

Integers

Wholenumbers

Whole numbers and their opposites.

Natural Numbers - Natural counting numbers.

1, 2, 3, 4 …

Whole Numbers - Natural counting numbers and zero.

0, 1, 2, 3 …

Integers -… -3, -2, -1, 0, 1, 2, 3 …

Integers, fractions, and decimals.Rational Numbers -

Ex: -0.76, -6/13, 0.08, 2/3

Rational Numbers

Name all the sets of numbers to which the givennumber belongs. Circle the most specific set.

1) 5

22) 3

3) 16

4) 0

5) 0.7

Integer, Rational

Rational

Rational

, Integer , RationalNaturals , Whole

, Integer , RationalWhole

Venn Diagram

Natural1, 2, 3...

Whole0

Integer11 5

Rational

6.7

59

0.8

327

Real Numbers

Remember… Rational numbers can be written as a fraction

or…as either a terminating or repeating decimal.

3 = 3.84 5 = 0.62

31.44 = 1.2

Classify the Following:•

– IrrationalIrrational

• -0.33333…– Rational (equals -⅓)Rational (equals -⅓)

23

Classify the Following:• 0.818811888111…

– Irrational (no end, no repetition)Irrational (no end, no repetition)

• 1⅔

– Rational (can be Rational (can be 55//3 3 ))

•

– Rational (equals 10 or Rational (equals 10 or 1010//1 1 ))

100

Rational v. Irrational – How alike?Rational v. Irrational – How alike?

• Subsets of Real numbersSubsets of Real numbers

• Can be negativeCan be negative

• Can be non-terminating Can be non-terminating (never end)(never end)

Rational v. Irrational – How different?Rational v. Irrational – How different?

• RRational:ational:– CAN be a fractionCAN be a fraction

– HAS a perfect HAS a perfect square rootsquare root

– Can be Can be terminating or terminating or repeating decimalsrepeating decimals

• Irrational:Irrational:– CANNOT be a CANNOT be a

fractionfraction

– Has NO perfect Has NO perfect square rootsquare root

– Can only be non-Can only be non-terminating, non-terminating, non-repeating decimalsrepeating decimals

A repeating decimal may not appear to repeat on a calculator, because calculators show a limited number of digits!

Caution!

Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as a fraction. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so is irrational.

Identify each root as rational or irrational.

1) 10

2) 25

3) 15

4) 49

6) 62

7) 81

8) 16

9) 99 irrational

irrational

irrational

rational

rationalrational

irrational

rational

5) 50 10) 121 rationalirrational

Decimal to Fraction: A skill Decimal to Fraction: A skill you need for this unit!you need for this unit!

• To change a decimal to a fraction, take the To change a decimal to a fraction, take the place value and simplify!place value and simplify!

• 0.5 means “5 tenths,” so start with 5/10

• Now simplify 5/10 to ½

• So… 0.5 = ½

Converting Fractions and DecimalsFraction Decimal

38

means 3 8

8 3.0000 3

2460

7

5640

5

400

0.375

To change a fraction to a decimal, take the top divided by the bottom, or numerator divided by the denominator.

Complete the table.Fraction Decimal

45 0.8

3100 0.03

720 0.35

7610 6.7

198

9.125

Repeating Decimals

Fraction Decimal13

means 1 3

3 1.0000 3

910

3

910

3

91

0.3...

0.33

Every rational number (fraction) either terminatesOR repeats when written as a decimal.

Repeating Decimals

Fraction Decimal5

11

means 5 11

11 5.000000 4

4460

5

5550

4

44

0.454...

0.454

60555044

54

6

0.45

Repeating Decimals

Fraction Decimal56

means 5 6

6 5.0000 8

4820

3

1820

3

182

0.83...

0.833

0.83

Rational Numberso CANCAN be made into a fraction be made into a fraction aa//bb, ,

where b where b ≠ 0.≠ 0.o A repeating OR terminating A repeating OR terminating

decimal.decimal.o 2/3 o o 0.798798798…

525 6.0

Irrational Numbers• CANNOTCANNOT be made into a fraction be made into a fraction

aa//bb, where b , where b ≠ 0.≠ 0.• A non-repeating AND non-A non-repeating AND non-

terminating decimal number.terminating decimal number.o πo o 0.313311333111…

5