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