Unit 1: Number Theory

1
Unit 1 Number Theory

• 5th Grade

2
Learning Goals

• Identify even and odd numbers.
• Draw arrays to model multiplication.
• Know multiplication facts.
• Use a divisibility test to determine if a number
  is divisible by another number.
• Find the factors of numbers.
• Identify prime and composite numbers.
• Find the prime factorizations of numbers.
• Rename numbers written in exponential notation.
• Understand how square numbers and their square
  roots are related.

3
Even and Odd Numbers

• An EVEN number is divisible into two equal whole
  amounts.
• An ODD number is not divisible into two equal
  whole amounts.

8 is even because it can be divided into two
equal whole amounts.
7 is odd because it cannot be divided into two
equal whole amounts.
4
Try it out!

• Which circle contains only odd numbers?

9, 5, 7, 4, 17, 23
3, 5, 8, 21, 29, 43
3, 9, 21, 43, 5, 15
5
Correct!

• Odd numbers are not evenly divisible by 2.

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6
Keep Trying!

• Odd numbers are not evenly divisible by 2.

9, 5, 7, 4, 17, 23
Neither one of these groups of numbers is all odd
because 4 and 8 are both even numbers.
3, 5, 8, 21, 29, 43
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7
Drawing Rectangular Arrays

• To draw arrays, think of the multiplication
  factors as the lengths of the sides and the
  product as the total amount.

12 x 1 12
4 x 3 12
6 x 2 12
3 x 4 12
2 x 6 12
1 x 12 12
8
Try it out!

• Which array represents the multiplication problem
  4 x 7 28?
• (Click on the circle)

9
Correct!

• An array is a complete rectangle with the number
  of rows and columns the same as the factors of
  the multiplication problem and the total number
  is equal to the product.

4 x 7 28
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10
Keep Trying!
4 x 7 28

• An array is a complete rectangle with the number
  of rows and columns the same as the factors of
  the multiplication problem and the total number
  is equal to the product.

Correct
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11
Know Your Facts!

• Here are some links to help develop fact fluency

http//www.multiplication.com/ http//www.school
express.com/funtime/math_generator/index.php
http//www.multiplication.com/interactive_games.
htm Find more on http//guest.portaportal.com/re
negar
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12
Use a Divisibility Test
13
Try It Out!

• The number 171 is divisible by

1, 3, 6, and 9
1, 3, and 9
1, 3, and 6
14
Correct!

• 171 is divisible by 1, 3, and 9 because
• All whole numbers are divisible by 1
• The digits added together are 9
• The digits added together (9) are divisible by 3

171 / 1 171 171 / 3 57 171 / 9 19
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15
Keep Trying!

• 171 is divisible by 1, 3, and 9 because
• All whole numbers are divisible by 1
• The digits added together are 9
• The digits added together (9) are divisible by 3

171 / 1 171 171 / 3 57 171 / 9 19
171 is not divisible by 6 because it is not
divisible by 2. (Its an odd number.)
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16
Find the Factors of Numbers

• A FACTOR is a number that is multiplied to equal
  a product. Products are divisible by their
  factors.

Factor (3)
Total (product) 15
5 x 3 15
product
factors
Factor (5)
15 / 3 5 15 / 5 3
17
Ways to Find Factors
Create arrays.
6
3
Factors of 6 are 1, 2, 3, 6
1
2
Divide.
You already know 1 and 6 are factors, divide by
each number (record only the ones that divide
evenly) and stop when you begin to see repeating
numbers.
6 / 6 1 6 / 1 6 6 / 2 3 6 / 3 2
stop
18
Try It Out!

• Find all of the factors of 48.

1, 48, 2, 4, 6, 8, 12, 24
1, 48, 2, 3, 4, 6, 8, 12, 16, 24
1, 48, 2, 3, 4, 6, 8, 12, 24
19
Correct!
48
Remember to record only numbers that divide
evenly!
48 / 48 1 48 / 1 48 48 / 2 24 48 / 3
16 48 / 4 12 48 / 6 8 48 / 8 6
If you chose to make arrays, you would have 5
different arrays.
Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24,
and 48
Stop
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20
Keep Trying!
48
Remember to divide sequentially (in order) and
record only numbers that divide evenly!
48 / 48 1 48 / 1 48 48 / 2 24 48 / 3
16 48 / 4 12 48 / 6 8 48 / 8 6
If you chose to make arrays, you would have 5
different arrays.
Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24,
and 48
Stop
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21
Identify Prime and Composite Numbers

• Prime numbers are numbers greater than one whose
  only factors are 1 and the number itself. There
  is only one array for prime numbers.
• Composite numbers are numbers greater than one
  who have more than 2 factors. There are multiple
  arrays for composite numbers.

22
Try It Out!

• Which array represents a prime number? (Click on
  the circle)

23
Correct!

• 3 is a prime number! Its only factors are 1 and
  3. It can only be made into one array.

These are really the same arrangement, just
rotated 90 degrees.
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24
Try Again!

• 3 is a prime number! Its only factors are 1 and
  3. It can only be made into one array.

These are really the same arrangement, just
rotated 90 degrees.
9, 6, and 4 are composite numbers because they
can be made into other arrays. They have more
than 2 factors.
incorrect
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25
Find the prime factorizations of numbers

• Finding PRIME FACTORIZATION is finding all of the
  prime factors of a number.

32
2 x 2 x 2 x 2 x 2
A Factor String is a string of factors that equal
a number. 32 2 x 2 x 8 A Prime Factorization
is the longest possible factor string. 32 2 x
2 x 2 x 2 x 2
These are all the prime factors of 32.
2
16
2
8
2
4
2
2
26
Try it out!

• What is the prime factorization of 45?

9, 5
3, 5
3, 3, 5
27
Correct!

• To find the prime factorization for 45, you need
  to break it up into factors.

45
5 x 3 x 3
5, 3, and 3 are all prime factors of 45.
5
9
3
3
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28
Keep Trying!

• To find the prime factorization for 45, you need
  to break it up into factors.

45
5 x 3 x 3
The answer is not 5, 3, because 5 x 3 15. In
order to be the prime factorization, all of the
factors multiplied together must equal the
beginning number. The answer is not 9, 5,
because 9 is not prime.
5, 3, and 3 are all prime factors of 45.
5
9
3
3
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29
Rename Numbers Written In Exponential Notation

• EXPONENTIAL NOTATION uses exponents to indicate
  an operation.

exponent
53 or 53 5 x 5 x 5 When you have an exponent,
it means to multiply the base number by itself
that many times. In this case, 5 is the base
number and 3 is the exponent.
42 4 x 4
base
42 16
30
Try It Out!

• Evaluate the following expression

32
9
6
8
31
Correct!

• 32 3 x 3
• 3 is the base number, 2 is the exponent. The
  exponent (2) means that you should multiply the
  base (3) by itself twice.
• 3 x 3 9

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32
Keep Trying!

• 32 3 x 3
• 3 is the base number, 2 is the exponent. The
  exponent (2) means that you should multiply the
  base (3) by itself twice.
• 3 x 3 9

Remember that the base is the number to multiply,
the exponent is the number of times to multiply
it!
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33
Square Numbers and Square Roots

• A number is square if it forms a square array.
  That is, one of the factors can be multiplied by
  itself to equal the product.

4
42 means 4 squared. It forms an array that is
4 x 4.
16 is a SQUARE NUMBER because it has a factor (4)
that can be multiplied by itself.
4
42 16
34
Square Roots

• Finding a SQUARE ROOT is simply undoing a
  square.
• v is the symbol for square root

You can also find a square number by listing its
factors.
9 1 x 9 9 3 x 3
9 32
v9 3
To find the square root of 9, you can arrange 9
pieces into a square (equal sides). The rows and
columns would both be 3, so 3 is the square root
of 9.
35
Try It Out!

• Which pair is correct?

62 12 v12 6
62 12 v36 6
62 36 v16 4
42 16 v12 6
36
Correct!

• 62 means 6 x 6. An array of 6 rows by 6 columns
  gives a product of 36 total.
• v16 means what number times itself is equal to
  16?. 4 x 4 16

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37
Keep Trying!

• 62 means 6 x 6. An array of 6 rows by 6 columns
  gives a product of 36 total.
• v16 means what number times itself is equal to
  16? 4 x 4 16

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