Unit 1 Number Theory (5th Grade)

Unit 1: Number Theory

5th Grade

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.

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.

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

Correct!

• Odd numbers are not evenly divisible by 2.

1

2Even

Odd 3 5 7 9 11 13 15 17

4 6 8 10 12 14 16 18

Click here to return to learning goals

• Odd numbers are not evenly divisible by 2.

1

2Even

Odd 3 5 7 9 11 13 15 17

4 6 8 10 12 14 16 18

Keep Trying!

9, 5, 7, 4, 17, 23

3, 5, 8, 21, 29, 43

Neither one of these groups of numbers is all odd because 4 and 8 are both even numbers.


Drawing Rectangular Arrays

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

3 x 4 = 122 x 6 = 12

4 x 3 = 126 x 2 = 12

1 x 12 = 12

12 x 1 = 12

Try it out!

• Which array represents the multiplication problem 4 x 7 = 28?

(Click on the circle)

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

4 rows

7 columns

28 total

Keep Trying!• 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.


2 rows

14 columns

28 total

5 rows

7 columns

35 total

4 x 7 = 28

4 rows

7 columns

28 total

Correct

Know Your Facts!

• Here are some links to help develop fact fluency:http://www.multiplication.com/

http://www.schoolexpress.com/funtime/math_generator/index.php

http://www.multiplication.com/interactive_games.htm

Find more on http://guest.portaportal.com/renegar


http://www.multiplication.com/









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http://guest.portaportal.com/renegar

Use a Divisibility Test

Divisible by Test2 Is an even number, ends in 0, 2, 4, 6, or 8

3 Sum of the digits is divisible by 3

5 Ends in 5 or 0

6 Divisible by both 2 and 3

9 Sum of all digits is 9 (or divisible by 9)

10 Ends in 0

11 Has double digits, pattern i.e. 33, 121, etc

Try It Out!

• The number 171 is divisible by:

1, 3, 6, and 9 1, 3, and 9 1, 3, and 6

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 3171 / 1 = 171

171 / 3 = 57

171 / 9 = 19


• 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

Keep Trying!

171 / 1 = 171

171 / 3 = 57

171 / 9 = 19

171 is not divisible by 6 because it is not divisible by 2. (It’s an odd number.)


Find the Factors of Numbers

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

5 x 3 = 15factors product

15 / 3 = 5

15 / 5 = 3

Factor (5)

Factor (3)

Total (product)

= 15

Ways to Find FactorsCreate arrays.

Factors of 6 are:

1, 2, 3, 6

Divide.

6 / 6 = 1

6 / 1 = 6

6 / 2 = 3

6 / 3 = 2

* 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.

stop

2

3

1

6

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, 24

1, 48, 2, 3, 4, 6, 8, 12, 16, 24

Correct!

48

48 / 48 = 1

48 / 1 = 48

48 / 2 = 24

48 / 3 = 16

48 / 4 = 12

48 / 6 = 8

48 / 8 = 6 Stop

Remember to record only numbers that divide evenly!

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


Keep Trying!

48

48 / 48 = 1

48 / 1 = 48

48 / 2 = 24

48 / 3 = 16

48 / 4 = 12

48 / 6 = 8

48 / 8 = 6 Stop

Remember to divide sequentially (in order) and record only numbers that divide evenly!

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


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.

Try It Out!

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

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.


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.

incorrect

9, 6, and 4 are composite numbers because they can be made into other arrays. They have more than 2 factors.


Find the prime factorizations of numbers

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

32

2 16

2 8

2 4

22

= 2 x 2 x 2 x 2 x 2

These are all the prime factors of 32.

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

Try it out!

• What is the prime factorization of 45?

9, 5 3, 5 3, 3, 5

Correct!

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


45

5 9

3 3

= 5 x 3 x 3

5, 3, and 3 are all prime factors of 45.

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

Keep Trying!

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.


45

5 9

3 3

= 5 x 3 x 3

5, 3, and 3 are all prime factors of 45.

Rename Numbers Written In Exponential Notation

• EXPONENTIAL NOTATION uses exponents to indicate an operation.

5^3 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

exponent

42 = 16

Try It Out!

• Evaluate the following expression:

32 =

9 6 8

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


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 = 9Remember that the base is the number to multiply, the exponent is the number of times to multiply it!


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.

42 means “4 squared”. It forms an array that is 4 x 4.

42 = 16

4

4

16 is a SQUARE NUMBER because it has a factor (4) that can be multiplied by itself.

Square Roots• Finding a SQUARE

ROOT is simply “undoing” a square.

• √ is the symbol for “square root”

√9 = 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.

You can also find a square number by listing its factors.

9 = 1 x 99 = 3 x 3 9 = 32

Try It Out!

• Which pair is correct?

62 = 12

√12 = 6

62 = 36

√16 = 4

62 = 12

√36 = 6

42 = 16

√12 = 6

Correct!• 62 means 6 x 6. An

array of 6 rows by 6 columns gives a product of 36 total.

• √16 means “what number times itself is equal to 16?”. 4 x 4 = 16


Keep Trying!

• 62 means 6 x 6. An array of 6 rows by 6 columns gives a product of 36 total.

• √16 means “what number times itself is equal to 16?” 4 x 4 = 16


Unit 1 Number Theory (5th Grade)

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