NUMBERS
----foundation of Mathematic
What we should do today:
Students will be given a video which contains all the things in this section we need to learn. While watching it, please not just laugh and play with others, you should use information you get from the video to complete this work sheet and hand it back at the end of this class.
Students will learn something about numbers in chapter 3 at this class, involving number sets, prime factors, greatest common factor(GCF), least common factor(LCM), square root and cube root.
Number Sets
To recognize each kind of numbers and express it easily and clearly, mathematicians make ______sets to do this job.
Numbers are divided up into several different sets:
______(N)
______
Often called “counting numbers” because we always use them to count.
______(W)
______
Includes all of the natural number set and has an extra Zero.
______(I or Z)
______or can be written as ______
Includes all of the natural number and whole number sets and has extra negative numbers.
______(Q)
______
Any number that can be written as a ratio (fraction)
Terminating decimal numbers, repeating decimal numbers
Includes all of the natural, whole, and integer number sets. .
______(Q’)
______
Any non-terminating, non-repeating decimal number
______(R)
All of the above number sets.
Examples for you to do:
1)Find out the irrational numbers among following:
1, 0, 1.365756, 3/5, 0.1235813…,
2)Determine the number sets of each number showing below:
- 0
- 3.6
- 3/4
- 5
- 1.142857142857…
- 1.23456789…
Factors , Prime, and Prime Factors
Every ______has at least one ______. The ______means the whole number elements of one whole number, which can ______together to it.
Example: 3 and 4 are factors of 12 for 3*4=12
Some whole numbers greater than one have only two distinct factors, which are one and itself, we call them ______.
Example: 7 is a prime number because its only factors are 1 and 7.
Numbers greater than one that aren’t prime numbers are called ______.
When the factors of a number a number are also prime, they are called ______.
Example: 8 has prime factors of 2*2*2 for 2 is prime. We call this the ______of 8.
The number 1 is not a prime number because it isn’t ______by any whole numbers other than itself.
The number 0 is not prime because it doesn’t have two ______factors.
Factor trees is a ______used to write the ______of a whole number.
Examples:
1)Write down all the factors of 24
1, 2, 3, 4, 6, 8, 12, 24.
2)Factor the number 18 in two different ways.
18 = 2*9
18 = 3*6
3)Write four examples of prime numbers.
23, 31, 53
4)Factor 54 into prime factors.
54 = 2 * 3^3
Examples for you to do:
1)Write down all the factors of 72 and factor it in three different ways, where must contain a prime factorization.
2)Write all the prime factors which are bigger than 20 and lower than 100.
GCF and LCM
The greatest common factor of two or more whole numbers is the ______whole number that is a ______of two or more numbers.
Example: The GCF of 16 and 36 is 4.
Techniques to finding the GCF of different numbers:
1)List all the factors of the numbers, choose the largest factors shared by them.
2)Factor the numbers into produces of power of prime factors. The GCF is the product of common powers with the smallest exponents associated with each other.
3)Divide the numbers by common prime factors until all the quotients do not have a common prime factor. In the method shown, the quotients are written under the dividends (divided numbers).
Examples:
1) Determine the GCF of 18 and 24
18: 1, 2, 3, 6, 9, 18.
24: 1, 2, 3, 4, 6, 8, 12, 24.
So the GCF is 6.
2) Determine the GCF of 24, 36, and 64
24= 2^3 * 3
36= 2^2 * 3^2
64= 2^6
So the GCF is 2^2 equals 4.
3) Determine the GCF of 12, 18, and 27
3 / 12 / 18 / 274 / 6 / 9
So the GCF is 3.
If numbers don’t have common prime factor, the product of common prim factors which are written in the ______is the GCF.
Examples for us to do:
1)Determine the GCF of 15 and 9, use 3 different ways.
2)Determine the GCF of 12, 8, and 36.
3)Determine the GCF of 12, 8, and 9.
The least common multiple of two or more whole number is the ______whole number that is a ______of two or more whole numbers.
Example: The LCM of 6 and 8 is 24
Techniques to finding the LCM of different numbers:
1)List the multiples of numbers, until a common multiple is found.
2)Factor each number into products of powers of prime factors. The LCM is the product of the common powers that have the largest exponents associated with them, along with any non-common powers.
3)Use the similar division techniques as we used for the GCF, but keep dividing until none of the numbers in a row have a common prime factor. The LCM is the product of the left column and the button row.
Examples:
1)Determine the LCM of 4 and 6
4: 4, 8, 12, 16, 24, 36
8: 8, 16, 24, 32, 40, 48
So the LCM is 24
2)Determine the LCM of 6, 8, and 18
6= 2 * 3
8= 2^3
18= 2 * 3 ^2
So the LCM is 2^3 * 3^2 equals 72
3)Determine the LCM of 9, 16, and 18
3 / 9 / 16 / 182 / 3 / 16 / 6
3 / 3 / 8 / 3
1 / 8 / 1
So the LCM is 2*3*3*8=145
Examples for us to do:
1)Determine the LCM of 12 and 14, use 3 different ways.
2)Determine the LCM of 4, 6, and 15.
Perfect Square and Perfect Cube
For whole numbers, the word ______in the term perfect ______means that the square can be written as a product of ______whole numbers.
Similarly, a perfect ____ can be written as a product of ______whole numbers.
Also, we can think a prefect square as the ____ of a square and a perfect cube as the ______of a cube with whole number dimensions in geometry.
We can use square tiles and small cubes to determine or show that whether a number is a perfect square or cube or not.
Examples:
1)Determine if the number 14 is a perfect square.
Solution: 3^2=9<14, 4^2=16>14, so 14 isn’t a perfect square.
2)Deter mine if the number 64 is a perfect cube.
Solution: 3^3=27<64, 4^3=64, so 64 is a perfect cube.
Examples for us to do:
1)Determine the number 16 is a perfect square or not, use square tiles.
2)Determine the number 16 is a perfect cube or not, use small cubes.
Square Roots and Cube Roots
The ______of a number is the two ______numbers which have the product of it, and the ______of a number is the three ______numbers which have the product of it.
The symbol that we use to show the operation of taking the positive square root is __, and the symbol for cube root is __.
The symbol __ represents the positive square root and the negative one is represented by the symbol __. So the symbol for both positive and negative square roots is ____.
The square root of a negative number is not a ______. However, cube roots have no these problem.
Chart of some Perfect Squares and Perfect Cubes needs to know
1=1^2 / 25=5^2 / 81=9^2 / 8=2^34=2^2 / 36=6^2 / 100=10^2 / 27=3^3
9=3^2 / 49=7^2 / 121=11^2 / 125=5^3
16=4^2 / 64=8^2 / 144=12^2 / 1000=10^3
Examples:
1)Determine the positive square root of 576
Solution: 20^2=400, 30^2=900, so the positive square root x must be between 20 and 30, and closer to 20. Since 4^2=16, 6^2=36, so x is 24.
2)Determine the cube root of 6869
Solution: 10^3=1000, 20^3=8000, so the cube root y must be between 10 and 20, and closer to 20. Since 9^3=9, so x is 19.
Examples for us to do:
1)Determine the negative square root of 900
2)Determine the cube root of 216
3)Determine the cube root of 2744