Arrays
Arrays are a visual way to demonstrate the multiplication of 2 whole numbers. Objects are drawn into rows and columns. To label an array, you write a number model. The number model is in this formula:
Number of Rows X Number of Columns.
Example:
x x x x
x x x x 5x4=20
x x x x
x x x x
x x x x
In the example above, there are 5 rows and 4 columns. The numbers 5 and 4 are called factors. Factors are two numbers multiplied together. The total amount or product in this example is 20. The product is the answer to a multiplication problem.
Click here to see a video explaining arrays,.
Factor Rainbows
Factors are numbers that, when multiplied, equal a product. In other words, in a multiplication problem, the numbers that are being multiplied together are the factors.
Creating factor rainbows is an easy way to identify all of the factors for a given number (or product).
When finding the factors of a number, there are 2 strategies you can use.
STRATEGY 1:
First, you could start with 1 and multiply it with a number that equals the factor. Then go to 2, 3, 4, and so on. For example:
Find all factors for the number 16:
1 x 16 = 16
2 x 8 = 16
3 x X =16 <---there are no whole number that when multiplied by 3 will equal 16
4 x 4 = 16
STOP! Once we find a factor that is multiplied by itself to get the product, we don't need to go any further, we would only find numbers that we have already identified as factors.
Now, we need to list all of the factors for 16 in order from LEAST to GREATEST. When writing the factors on paper, you connect the factor pairs with large arches, which will look like a rainbow. When listing a factor that is multiplied by itself, you only need to write the number once.
1, 2, 3, 4, 8, 16
STRATEGY 2:
Instead multiplying up, you could divide the product starting with 1 then continuing on to 2, 3, 4, and so on.
Find all factors for 2
16/1 = 16
16/2 =
16/3 = X <---16/3=5.3333 which is not a whole number.
16/4 = 4
Again, we need to list all of the factors for 16 in order from LEAST to GREATEST. When writing the factors on paper, you connect the factor pairs with large arches, which will look like a rainbow. When listing a factor that is multiplied by itself, you only need to write the number once.
1, 2, 3, 4, 8, 16
Creating factor rainbows is an easy way to identify all of the factors for a given number (or product).
When finding the factors of a number, there are 2 strategies you can use.
STRATEGY 1:
First, you could start with 1 and multiply it with a number that equals the factor. Then go to 2, 3, 4, and so on. For example:
Find all factors for the number 16:
1 x 16 = 16
2 x 8 = 16
3 x X =16 <---there are no whole number that when multiplied by 3 will equal 16
4 x 4 = 16
STOP! Once we find a factor that is multiplied by itself to get the product, we don't need to go any further, we would only find numbers that we have already identified as factors.
Now, we need to list all of the factors for 16 in order from LEAST to GREATEST. When writing the factors on paper, you connect the factor pairs with large arches, which will look like a rainbow. When listing a factor that is multiplied by itself, you only need to write the number once.
1, 2, 3, 4, 8, 16
STRATEGY 2:
Instead multiplying up, you could divide the product starting with 1 then continuing on to 2, 3, 4, and so on.
Find all factors for 2
16/1 = 16
16/2 =
16/3 = X <---16/3=5.3333 which is not a whole number.
16/4 = 4
Again, we need to list all of the factors for 16 in order from LEAST to GREATEST. When writing the factors on paper, you connect the factor pairs with large arches, which will look like a rainbow. When listing a factor that is multiplied by itself, you only need to write the number once.
1, 2, 3, 4, 8, 16
Divisibility Rules
There are tricks to help you determine if a number is divisible by 2, 3, 4, 5, 6, 9, and 10. Use these quick tricks instead of dividing to find factors.
Rules and Examples
Easy, "Look-at-the-Number" Divisibility Rules
2- any even number (a number ending in 0, 2, 4, 6, 8) is divisible by 2
example - 746 is divisible by 2 because it ends in 6
4 - if the numbers in the tens and ones place is divisible by 4, then the whole number is divisible by 4.
example - 716......16 is divisible by 4, therefore 716 is divisible by 4.
5 - any number ending in 5 or 0 is divisible by 5
example - 630 is divisible by 5 because it ends in 0
example - 965 is divisible by 5 because it ends in 5
10 - any number ending in 0 is divisible by 10
example - 619,950 is divisible by 10 because it ends in 0
"Work-for-it" Divisibility Rules
These rules involve some work; you cannot just look at the number to determine divisibility.
3 - if the sum of the digits is divisible by 3, then the original number is divisible by 3
example - 252....2 + 5 +2 = 9 is divisible by 3, therefore, 252 is divisible by 3.
6 - if a number is divisible by BOTH 2 AND 3, then the number is divisible by 6.
example - 138 - 8 is even, so it is divisible by 2.....1 + 3 + 8 = 12 is divisible by 3. Since 138 is divisible by 2 and 3, then it is divisible by 6.
9 - if the sum of the digits is divisible by 9, then the original number is divisible by 9.
example - 963.....9 + 6 + 3 = 18 is divisible by 9, therefore 963 is divisible by 9.
Rules and Examples
Easy, "Look-at-the-Number" Divisibility Rules
2- any even number (a number ending in 0, 2, 4, 6, 8) is divisible by 2
example - 746 is divisible by 2 because it ends in 6
4 - if the numbers in the tens and ones place is divisible by 4, then the whole number is divisible by 4.
example - 716......16 is divisible by 4, therefore 716 is divisible by 4.
5 - any number ending in 5 or 0 is divisible by 5
example - 630 is divisible by 5 because it ends in 0
example - 965 is divisible by 5 because it ends in 5
10 - any number ending in 0 is divisible by 10
example - 619,950 is divisible by 10 because it ends in 0
"Work-for-it" Divisibility Rules
These rules involve some work; you cannot just look at the number to determine divisibility.
3 - if the sum of the digits is divisible by 3, then the original number is divisible by 3
example - 252....2 + 5 +2 = 9 is divisible by 3, therefore, 252 is divisible by 3.
6 - if a number is divisible by BOTH 2 AND 3, then the number is divisible by 6.
example - 138 - 8 is even, so it is divisible by 2.....1 + 3 + 8 = 12 is divisible by 3. Since 138 is divisible by 2 and 3, then it is divisible by 6.
9 - if the sum of the digits is divisible by 9, then the original number is divisible by 9.
example - 963.....9 + 6 + 3 = 18 is divisible by 9, therefore 963 is divisible by 9.
Place Value
- Each place directly to the left is 10x larger than the place to the right.
- Each place directly to the right is 1/10 the size of the place directly to the left.
WHOLE NUMBERS
| Millions Period | Thousands Period | Ones Period |
| hundreds | thousands | ones| hundreds | thousands | ones| hundreds | thousands | ones|
Decimals
| ones | tenths | hundredths | thousandths |
Comparing Numbers
> greater than
< less than
= equal to
In order to compare numbers, stack them and determine which number is greater or less than by looking at the first digit that is different than the other.
Example:
345,687,201
345,878,102
The numbers 345 are the same, but 6 is less than 8, therefore 345,687,201 < 345,878,102. You could also say 345,878,102 > 345,687,201
Click here for place value games.
Square Numbers
Any number that has a factor that is multiplied by itself, is a square number. The product is the whole number.
1x1= 1
2x2= 4
3x3 = 9
4x4 = 16
5x5 = 25
6x6 = 36
7x7 = 49
8x8 = 64
9x9 = 81
and so on.
1x1= 1
2x2= 4
3x3 = 9
4x4 = 16
5x5 = 25
6x6 = 36
7x7 = 49
8x8 = 64
9x9 = 81
and so on.
