Download Unit 1 Overview: Number Theory ARRAYS SQUARE NUMBERS

Unit 1 Overview: Number Theory
ARRAYS
arrays must have the same number of objects in each row
arrays must have the same number of objects in each column
this is an array for 10:
x
x
these are NOT arrays for 10: x
x
x
x
x
x
x
x
x
x
x
x xx
xx
row x column = product
factor x factor = product
(or multiple)
x
x
or
x x x x this is an
x
x outline
this is too uneven
x x x x
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SQUARE NUMBERS
Sometimes you can make a square array. You can only make square arrays with certain numbers.
The number of rows and the number in each row must be the same.
1
4
9
16
x
x x
x x x
x x x x
x x
x x x
x x x x
x x x
x x x x
x x x x
The shorthand for those multiplication problems uses an exponent.
2
4 = 2 x 2 = 2
read it as ‘2 squared’ or ‘2 to the second power’
2
9 = 3 x 3 = 3
read it as ‘3 squared’ or ‘3 to the second power’
2
16 = 4 x 4 = 4
read it as ‘4 squared’ or ‘4 to the second power’
Since you can square a number, you can also ‘unsquare’ it. When you do that, you find the
square root of the number. It is the number in a row or column of the square array. Or, the number
you multiply by itself to get the root.
To find the square root of 9,
think
OR
make/picture a square array of 9 x’s
What number times itself equals 9?
How many in a row or column?
3x3=9
x x x
x x x
x x x
either way you think about it, the square root of 9 is 3
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FACTORS
3 x 2 =6
3 and 2 are factors of 6
6 is the product of 3 and 2
1 x 6 = 6
1 and 6 are factors of 6
6 is the product of 1 and 6
to list all the factors of a number: list the factor pairs in order, or use Factor Rainbows
the factors of 24 are
the factors of 23 are
(1, 2, 3, 4, 6, 8, 12, 24)
(1, 23)
DIVISIBILITY RULES
or
(1, 24, 2, 12, 3, 8, 4, 6)
the factors of 2 are (1, 2)
any even number is ÷ 2
a number ending in 0 or 5 is ÷ 5
a number ending in 0 is ÷ 10
for ÷ 3, add the digits -- if the sum is ÷ 3, so is the number
for ÷ 9, add the digits -- if the sum is ÷ 9, so is the number
any even number that is also ÷3 is ÷6 (2 & 3 are factors of 6)
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PRIME OR COMPOSITE? EVEN OR ODD?
24 and 27 both have more than 2 factors, so they are composite numbers
23 and 11 have 2 and only 2 factors, one and the number itself, they are prime numbers
odd numbers end in 1, 3, 5, 7, or 9
even numbers end in 0, 2, 4, 6, or 8
23 and 27 are odd numbers
24 and 2 are even numbers
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FACTOR STRINGS AND PRIME FACTORIZATION
a factor string is a name for a number written as a multiplication number sentence (no 1’s)
factor strings can be short or long
some factor strings for 36 are
2 x 18
2x3x6
4x3x3
some factor strings for 28 are
2 x 14
4x7
2x2x7
the longest string (which has only prime numbers) is the prime factorization of that number
the prime factorization of 36 is
3x3x4
the prime factorization of 28 is
2x2x7
finding the prime factorization is easy
*think of any fact for the number
*circle any prime numbers, write a
new fact for the composite numbers
for 50
for 48
5 x 10
6x8
5x2x5
2x3x2x4
2x3x2x2x2
*once all the numbers are prime, you’re done 5 x 2 x 5
2x3x2x2x2
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PLACE VALUE
Know large places to one million
million, hundred
ten
thousands, hundreds
tens
ones
thoudsands
thousands
4,
8
5
9,
2
7
3
four million, eight hundred fifty nine thousand, two hundred seventy-three