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1.2 Rectangular Arrays
Make a rectangular array using 6 dots. Label the array by writing a number model
next to the array. Can you draw more arrays using 6 dots? Label each array.
The arrays you drew show the turn-around rule for multiplication. Give 1 example of
the turn-around rule.
____________________
____________________
Give another example of the turn-around rule.
____________________
____________________
1.3 Factors
In a number model such as 3 * 5 = 15, the 3 and 5 are called factors. The 15 is called a
product of 3 and 5. The factors 3 and 5 are a factor pair for the number 15. Can you
name another factor pair for 15?
_____ * _____ = _____
(Label the factors and product.)
Are there other whole-number factor pairs for 15? YES
NO
Make arrays for each factor pair of 18. Show your work below. Don’t forget to label
your arrays with a number model!
List the whole-number factor pairs for
18.________________________________________________________
1.4 The Factor Captor Game
Why is 4 a factor of
36?_____________________________________________________________________
_
Why is 5 not a factor of
36?___________________________________________________________________
When you are dividing 42 divided by 7, you can think: 7 times what number is 42? (6)
Since you get a whole-number answer (6), you can say: 7 is a factor of 42 because 42 is
divisible by 7.
A number is divisible by another number if the result of the division is a whole number,
with a remainder of zero. For example, 28 is divisible by 7 because 28 divided by 7 is 4,
with a remainder of zero. Since 28 is divisible by 7, 7 is a factor of 28.
Is 54 divisible by 9?_____ Why?_____________________________
Is 25 divisible by 6?_____ Why?_____________________________
Is 8 a factor of 48?______ Why?_____________________________
Is 4 a factor of 30?______ Why?_____________________________
1.5 Divisibility
A whole number is divisible by a whole number if the remainder in the division is zero.
The answer to the division problem (quotient) must be a whole number. If the remainder
is not zero, then the first number is not divisible by the second number.
30 is divisible by 5 because 30 / 5 = 6
Remainder 2 or 7.5
6 is a whole number, so
a decimal in the
30 is divisible by 5
30 is not divisible by 4 because 30 / 4 = 7
Since there is a remainder in the answer (or
answer), this means 30 is not divisible by 4.
If checking for divisibility on a calculator, if you see a decimal in the answer to a
division problem then first number is not divisible by the second number.
(27 divided by 5 = 5.4  since the answer has a decimal, 27 is not divisible by 27.
There are different tests you can do to determine if a number is divisible by another
number:
All numbers are divisible by _____.
A number is divisible by 2 if it ends in a _____ _____ _____ _____ _____.
A number is divisible by 3 if the sum of its digits is divisible by 3.
Example: 246 is divisible by 3 because 2 + 4 + 6 = 12, and 12 is divisible by 3 
12 / 3 = 4
A number is divisible by 6 if it is divisible by both 2 and 3.
Example: 246 is divisible by 6 because it is divisible by both 2 and 3
(it ends in a 6, so it is divisible by 2 and 2 + 4 + 6 = 12, and 12 is divisible
by 3  12 / 3 = 4)
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example: 51,372 is divisible by 9 because 5 + 1 + 3 + 7 + 2 = 18, and 18 is
divisible by 9  18 / 9 = 2
A number is divisible by 5 if it ends in a _____ or _____.
A number is divisible by 10 if it ends in a _____.
Try using the divisibility test for 3. See if these numbers are divisible by 3:
Is 237 divisible by 3?_____
Why?_____________________________________________________________
Is 415 divisible by 3?_____
Why?_____________________________________________________________
Write a number that is divisible by 3: __________
Why?_____________________________________
1.6 Prime and Composite Numbers
__________ Numbers
_______________ Numbers
Definition:_______________________________
Definition:_________________________________
Write the factor pairs for each number and then make a factor rainbow number.
2-dot arrays
4-dot arrays
5-dot arrays
10-dot arrays
11-dot arrays
16-dot arrays
1.7 Square Numbers
The number 16 can make an array that has the same number of rows and columns. (4 by
4 array)
Draw the array below and make a square around it.
Because the array is shaped like a square, it is called a _______________
_______________.
Therefore, the number 16 is called a _______________ _______________.
Draw the square array for the number 1
the number 4
Don’t forget to draw a square around it!
around it!
Draw the next square array.
have?____
Draw the square array for
Don’t forget to draw a square
How many rows and columns does it
What is the square number?____
Draw the next square array.
have?____
How many rows and columns does it
What is the square number?____
1.8 Unsquaring Numbers
When you ask yourself: What number, multiplied by itself, is equal to 81? (9) you are
“unsquaring” a number.
Use your calculators to answer this questions: What number, multiplied by itself, is equal
to 289?__________
Do not use the “square root” key.
Another name for “unsquaring” a number is called finding the _______________
__________ of the number.
Since 8 * 8 = 64, the square root of 64 is 8.
1.9 Factor Strings and Prime Factorizations
A _______________ _______________ is a name for a number written as a product of
at least two factors that are greater than 1. In a factor string, the number 1 may not be
used as a factor.
Example: A factor string for the number 24 is 2 * 3 * 4.
You could also write the factor string in a different order, such as 2 * 4 * 3 or 3
* 2 * 4 or 3 * 4 * 2
What are two other ways you could write the factor string for
24?_______________ _______________
The length of the factor string for each of these examples is 3. You could write other
factor strings for the number 24 using different lengths.
Fill in the grid below, recording all possible factor strings for the number 24.
Number
Factor String
Length
24
24
24
24
24
24
Fill in the grid below, recording all possible factor strings for the number 30.
Number
Factor String
Length
30
30
30
Fill in the grid below, recording all possible factor strings for the number 50.
Number
Factor String
Length
50
50
50
Fill in the grid below, recording all possible factor strings for the number 54.
Number
Factor String
Length
54
54
54
54
54
54
Fill in the grid below, recording all possible factor strings for the number 72.
Number
Factor String
Length
72
72
72
72
72
72
72
72
72
72
72
72
72
72
72
Fill in the grid below, recording all possible factor strings for the number 36.
Number
Factor String
Length
36
36
36
36
36
36
36
36
What kind of numbers make up the longest possible factor string for a
number?__________ ______________
The longest factor string for a number is called the __________
_________________________ of a number.
Example: The prime factorization of 24 is 2 * 2 * 2 * 3
The easiest way to find the prime factorization of a number is to do a factor tree.
Example:
24
3
*
8
2
*
4
2
*
2
Circle the prime numbers. The numbers you circle will be the prime factorization of
the number 24.
Make a factor tree for the number 20. Write out the prime factorization for the number
20:__________
20