5.2 Irrational Numbers

i-can

By memory, write the decimal form for each Benchmark Fraction:

Write down this problem on your

COMMUNICATORBe prepared to share your work with the

class.

Page 48 of your INB

estimate where irrational numbers are placed on a number line by determining which two whole numbers is in between.

1) = _____ 2) = _____ 3) = _____

4) = _____ 5) = _____ 6) = _____ 7) = _____ 8) = _____ 9) = _____

0.50.25 0.75

0. 0. 0.125

0.375 0.625 0.875

ITEMS of BUSINESS

• Rework is due TODAY• Make plans to retake the test if you need to. - Practice Test completed - REWORK completed - 3x5 card made - schedule a time to retake* Last day to retake will be next Tuesday, DEC 9th.

Homework Help

Have your homework out on your desk.

RE-Hey You. Try this CHALLENGE question:

Convert this Decimal to a fraction:3.5787878… (must show work)

1000x = 3578.7878… -10x = -35.7878… 990x = 3543 x = =

Assign: x = 3.57878…

Do you see a pattern?Last time we converted repeating decimals to fractions, such as these examples…

0.555… 10x = 5.555… - x =-0.555… 9x = 5 x =

Assign: x = 0.555…

0.444… 10x = 4.444… - x =-0.444… 9x = 4 x =

Assign: x = 0.444…

0.151515… 100x = 15.1515… - x = -0.1515… 99x = 15 x = =

Assign: x = 0.151515…

This Shortcut only works

when the repeating numbers are right after

the decimal.0.1333…

100x = 13.333… -10x = -1.333… 90x = 12 x = =

Assign: x = 0.1333…

So this type of equationdoes not follow the

pattern.

Do you see a pattern?

Get into your 4-Square Groups

i-canestimate where irrational numbersare placed on a number line by determining which two wholenumbers is in between.

Page 48-48 of your INB

Perfect SquaresPerfect are squares of Natural numbers

Squares

They LITERALLY describe the

3

3

3 3 = = 9

3

311

2

2

4

4

1 4 9

16 1, 4, 9 & 16are

PERFECTSQUARES

Are theremore?

Perfect Squares

=

=

==

=

=

==

=

=

=

=

MEMORIZE THEM1491625364964

81100121144

Are these Perfect Squares?

Thumbs Up

PerfectSquares

Thumbs down

Not PerfectSquares3615648110100

SQUARE ROOTS

√❑Taking the square root of aperfect square

will give you thedimension of

one side of the square

3

39

√9= 3

2

Square Roots UNDO Squares

= √323 2

INVERSE OPERATIONS𝑥2 √𝑥

Let’s try a few

=

= = 7= 5√52

√72

=

=

√22 = 2√122 = 12

= √12 = 1

Square Roots jkkkk Foldable: Highlight the Perfect Squares

Class Activity

Where can I find on the

Number line?(without a calculator)

√20√9 √16 √25

i-can estimate where irrational numbers are placed on a number line by determining which two whole numbers is in between.

Between which two Integers would you find…

√13is in between 3 & 4

√99 is in between 9 & 10

Square Roots jkkkk

Square Roots UNDO Squares

√❑1) 36INVERSE

OPERATIONS𝑥2 √𝑥We can use this inverse operation to solve equations with exponents.

√❑6 √❑2) 121√❑11 √❑3) √❑𝑥=√5

WHICH IS MORE ACCURATE? or 2.2 (decimal approximation of is 2.2360679…𝑥=√5

𝑎𝑛𝑠𝑤𝑒𝑟 𝑖𝑠 h𝑒𝑖𝑡 𝑒𝑟 𝑥=√5𝑜𝑟 𝑥 ≈2.2

Let’s try a few more

2¿ A=100 𝑖𝑛2 , 𝑓𝑖𝑛𝑑𝑠 = 10 in

3¿ A=6.25 𝑓𝑡 2, 𝑓𝑖𝑛𝑑𝑠 = 2.5 ft

4¿ A=267.5𝑚2 , 𝑓𝑖𝑛𝑑𝑠 16.4 m

1¿ h𝑇 𝑒 𝐴𝑟𝑒𝑎𝑜𝑓 𝑎𝑆𝑞𝑢𝑎𝑟𝑒𝑖𝑠16𝑚2 , h𝑤 𝑎𝑡 𝑖𝑠 h𝑡 𝑒𝑠𝑖𝑑𝑒 h𝑙𝑒𝑛𝑔𝑡 ?

s = 4ms =

𝑠=√16𝑚2

𝐴=𝑠2

16 = s

Perfect CubesPerfect are cubes of Natural numbers

Cubes

They LITERALLY describe the

3

3

3 3 3 = = 273

Perfect Cubes

=

=

=

=

=

MEMORIZE THEM1 64 8 12527

CUBE ROOTS

√❑Taking the

Cube root of aperfect cube

will give you thedimension ofone edge of

the cube

3 3

3√27 = 3

33

27

Cube Roots UNDO Cubes

= √333 33

INVERSE OPERATIONS𝑥3 3√𝑥

Let’s try a few

=

=

=

= 1= 2= 4

3√23

3√13

3√ 43

Cube Roots Foldable: Highlight the Perfect Cube

Between which two Integers would you find…

3√7is in between 1 & 2

3√80 is in between 4 & 5

Cube Roots UNDO Squares

3√❑1) 8INVERSE

OPERATIONS𝑥3 3√𝑥We can use this inverse operation to solve equations with exponents.

3√❑2 3√❑2) 1253√❑5 3√❑3) 3√❑𝑥=3√7

WHICH IS MORE ACCURATE? or 1.9 (decimal approximation of : 1.912931183…)𝑥=3√7

𝑎𝑛𝑠𝑤𝑒𝑟 𝑖𝑠 h𝑒𝑖𝑡 𝑒𝑟 𝑥=3√7𝑜𝑟 𝑥≈1.9

Let’s try a few more

2¿𝑉=512 𝑖𝑛3 , 𝑓𝑖𝑛𝑑𝑠 = 8 in

3¿𝑉=0.216 𝑓𝑡3 , 𝑓𝑖𝑛𝑑𝑠

4¿𝑉=137.8𝑚3 , 𝑓𝑖𝑛𝑑𝑠

1¿ h𝑇 𝑒𝑣𝑜𝑙𝑢𝑚𝑒𝑜𝑓 𝑎𝐶𝑢𝑏𝑒𝑖𝑠125𝑚3 , h𝑤 𝑎𝑡 𝑖𝑠 h𝑡 𝑒 𝑠𝑖𝑑𝑒 h𝑙𝑒𝑛𝑔𝑡 ?

s = 5ms =𝑠=3√125𝑚3

𝑉=𝑠3

= s

= 0.6 ft

5.2 m

125

Is 625 a “perfect” 4th?

Is 3125 a “perfect” 5th?

How far up/down do you think we could take this whole “perfect” thing?

If 32 exists, is it reasonable to think that 3-2 exists?

Notice that you can have a negative in front of a root, like -. It’s just another way of saying -4.

What is 54? 55? ? ?

Are these numbers Rational?(Can they be written as a Fraction?)

Thumbs Up

Rational

Thumbs down

Not Rational

√16√813√273√1250.1250.254897…0.333…Rational

√ 𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑠𝑞𝑢𝑎𝑟𝑒𝑠3√𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑐𝑢𝑏𝑒𝑠

All numbers that can be written in the form .

Examples:

Decimals that terminateor repeat.

Examples: 0.356 and 0.555…

RATIONALNUMBERS

Examples: √ 𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑠𝑞𝑢𝑎𝑟𝑒𝑠

3√𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑐𝑢𝑏𝑒𝑠

Numbers that are NOT rational are Irrational.

Irrational numbers include ,

The decimal expansion of irrational numbers continue forever without any repeating pattern.

𝜋 √2

√203√10

𝜋  5

2.54893…√𝑛𝑜𝑛𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑠𝑞𝑢𝑎𝑟𝑒𝑠

3√𝑛𝑜𝑛𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑐𝑢𝑏𝑒𝑠

RATIONALNUMBERSAll numbers that can be written in the form .

Examples:

Decimals that terminateor repeat.

Examples: 0.356 and 0.555…

Examples:

IRRATIONALNUMBERS

Real numbers that areNOT RATIONAL

Decimals that go on forever and never repeat.

Examples: , & 0.987556…

Examples: -

The

CRAZY

Ones!

√ 𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑠𝑞𝑢𝑎𝑟𝑒𝑠3√𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑐𝑢𝑏𝑒𝑠 √𝑛𝑜𝑛𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑠𝑞𝑢𝑎𝑟𝑒𝑠

3√𝑛𝑜𝑛𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑐𝑢𝑏𝑒𝑠

Worksheet5.2

IrrationalNumbers