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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
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