1.6 Prime and Composite Numbers

1.6 Prime and Composite Numbers September 3, 2013

Math Message 1.6 • Please complete the following Math Message in your Math notebooks and have your answers ready to share: Draw ALL possible rectangular arrays for these numbers: 2, 4, 5, 10, 11, and 16

What exactly are prime numbers? • A prime number has exactly two factors- 1 and the number itself • Important to know: the number 1 is considered neither prime nor composite • Let’s look at some examples…. 2  1 and 2 17  17 and 1 23  1 and 23

What are composite numbers? • A composite number has more than two factors • Some examples of composite numbers:

1.7 Square Numbers September 4, 2013

Math Message 1.7 • Use coins, M & M candy, or any other items that resemble counters. • Try and make a rectangular array with an equal number of rows and columns for each of the following numbers: • 14 • 16 • 18 Which numbers make this kind of array? Show your work for each array.

Answer to the Math Message: • Of the three numbers, 16 is the only one that can be represented equally 4 x 4 array * * * * * * * * * * * * * * * *

How is a 4 x 4 array similar to a square? • When an array has the same number of rows and columns, it is shaped like a square and is called a square array • The number it represents is called a square number Since 16 can be represented by a square array, the number 16 is a square number

Finding other Square Numbers • How many objects are in the array? 9 • How many rows are in the array set? 3 • How many columns are in the array set? 3 3-by-3 array

Square Numbers • Any square number can be written as the product of a number multiplied by itself • 1 x 1 = 1 • 2 x 2 = 4 • 3 x 3 = 9 • 4 x 4 = 16 A shorthand way of writing square numbers looks like this: 9 = 3 x 3 = 32 Let’s work on MJ pg. 20

What are exponents? **You can read 32 as “3 times 3, 3 squared, or 3 to the second power” • The raised 2 is called an exponent. It tells you that 3 is used as a factor 2 times 32 = 3 x 3 = 9 • Exponential Notation- numbers that are written with an exponent ex. 32

What is the difference between doubling a number and squaring a number? • What do you get when you double 10? 20 = 10 +10 or 10 x 2 • What do you get when you square 10? 10 * 10 = 100 or 102

1.8 Unsquaring Numbers September 5, 2013

1.8 Math Message • Find the numbers that make these statements true. Write out and complete each statement in your Math notebook. 1) ______ * ______ = 4 2) ______ 2 = 81

What numbers could the blank spaces in these problems represent? • Answers to the Math Message: • The factors of 1 and 4, or 2 • The factor 9 • Now, let’s replace the blank spaces with the letter (the variable), N. What number is being represented by the variable, N? N * N = 4 N = 2

What do you know about variables? • A variable can only represent one number if the number sentence is true. • For example: m2 = 81 m squared equals 81 m * m = 81 What number is being represented by the variable m to make the number sentence true? m = 9

“Unsquaring” Numbers • When “unsquaring” a number, we need to “undo” the operation • If you square a number, you are multiplying it by itself to get the product 4 * 4 = p • When you are given the product, you will need to “undo” the multiplication to identify the number that was squared n * n = 16

Problems we’ll complete in class.. • What number, multiplied by itself, is equal to 289? • What strategies are you using? • Let’s practice “unsquaring numbers” • 196 • 10,000 • 7,225 • 441

Finding the Square Root of Numbers • When you “unsquare” a number, you have found the square root of the number What number squared is 64? 8 So, what is 64 squared? 8, because 8 * 8 = 64 What is the square root of 64? 8

Testing our results using a calculator • When using a calculator to find the square root you will need to know the following: 1) If the display shows a whole number, then the original number is a square number. *For example: 576 is a square number because using the square-root key displays a whole number, 24 2) If the display shows a decimal, then the original number is not a square number. * For example: 794 is not a square number because using the square-root key displays a decimal--- 28.178006 (rounded to 6 decimal places)

1.9 Factor Strings and Prime Factorizations September 6, 2013

1.9 Math Message • 8 + 8 and 4 * 4 are two names for the number 16. In your Math notebook, write at least five other names for 16. Brainstorm the different ways and be prepared to share your answers

What are Factor Strings? • A factor string is a multiplication expression that has at least two factors that are greater than 1. *In a factor string, the number 1 may NOT be used as a factor Example of a factor string for the number, 24: 24 is 2 * 3 * 4 • In the factor string, you see three factors (2, 3, 4), this is called, the length of the factor string *Now, let’s find other factor strings for 24…

Let’s practice…(in class) • Find the factor string for 7… • What type of number is 7? A prime number • The number 1 may not be used in a factor string, so there are no factor strings for prime numbers 3) Let’s find all possible factor strings for 36… • Is 2 a factor of 36? Yes, 36 = 2 * 18 • Is 2 a factor of 18? Yes, 36 = 2 * 2 * 9 • Is 2 a factor of 9? No • Is 3 a factor of 9? Yes, 36 = 2 * 2 * 3 * 3

Prime Factorization • What kind of numbers are the factors in the longest possible factor string for any number? Prime numbers • The longest factor string for a number is called the prime factorization of the number For example: The prime factorization of 24 is… 2 * 2 * 2 * 3

Using Factor Trees to find all the prime factors • We will practice creating a Factor Tree for 36 36 6 * 6 3 * 2 3 * 2 36 = 2 * 2 * 3 * 3 36 = 22 * 32

1.6 Prime and Composite Numbers