Warm Up – copy these into your notes on a new notes page!!! Divisibility Rules: A quick way to know if numbers are divisible by another number. 2: A number is divisible by 2 if… 3: A number is divisible by 3 if… 4: A number is divisible by 4 if… 5: A number is divisible by 5 if… 6: A number is divisible by 6 if… 8: A number is divisible by 8 if… 9: A number is divisible by 9 if… 10: A number is divisible by 10 if…

A number is divisible by 2 if the last digit of the number is 0, 2, 4, 6, or 8. Ex: 28, 536, 974

A number is divisible by 3 if the sum of the digits is divisible by 3. Ex: 12 » 1 + 2 = 3 96 » 9 + 6 = 15 945 » 9 + 4 + 5 = 18

A number is divisible by 4 if the last two digits create a number that is divisible by 4 or if the last two digits are 00. Ex: 348 » 48  4 = 12 328 » 28  4 = 7 500 (500  4 = 125)

A number is divisible by 5 if the last digit is a 0 or a 5. Ex: 45, 120, 935

A number is divisible by 6 if the number is divisible by 2 (even) and 3 (sum of digits divisible by 3). Ex: 846 » 8 + 4 + 6 = 18 522 » 5 + 2 + 2 = 9 1, 356 » 1 + 3 + 5 + 6 = 15

A number is divisible by 9 if the sum of the digits is divisible by 9. Ex: 81 » 8 + 1 = 9 945 » 9 + 4 + 5 = 18 7,578 » 7 + 5 + 7 + 8 = 27

A number is divisible by 10 if the last digit is a 0. Ex: 80, 470, 990

Warm Up Circle the factors of the numbers. Cross out numbers that are not factors. 3475 2 3 4 5 6 9 10 82 2 3 4 5 6 9 10 960 2 3 4 5 6 9 10 27 2 3 4 5 6 9 10

Prime Time Definitions Divisible: meaning that a number can divide evenly into another number Ex: 12 is divisible by 2 2. Product: the answer to a multiplication problem 3. Multiple: the product of a whole number and another whole number Ex: 4x3 is 12 so 12 is a multiple of 3 4. Least Common Multiple: The smallest number that is a multiple of two numbers. Ex: 12 is the LCM of 3 and 4.

Prime Time Definitions 5. Factor: one of two or more numbers that are multiplied to get a product Ex: 2 x 5 = 10 so 2 and 5 are factors of 10 6. Greatest Common Factor: the biggest factor that two or more numbers share in common 7. Prime Numbers: A number with only two factors: 1 and the number itself. Ex: the factors of 11 are 1 and 11. 8. Composite Numbers: A whole number with factors other than itself and 1. Ex: 4, 24, and 30 are composite numbers because they have many factors.

Prime Time Definitions 9. Square Number: The product of a number with itself. Ex: 3 x 3 = 9, 5 x 5 = 25, 6 x 6 = 36. 10. Exponent: The small raised number that tells you how many times to multiply a base (or factor) times itself. 11. Prime Factorization: The longest factor string for a number that is made up of all prime numbers.

The Product Game 1 2 3 4 5 6 7 8 9 10 12 14 15 16 18 20 21 24 25 27 28 30 32 35 36 40 42 45 48 49 54 56 63 64 72 81 Factors: 1 2 3 4 5 6 7 8 9 http://illuminations.nctm.org/ActivityDetail.aspx?ID=29

WARM UP 9-8-10 Follow Up Questions – Product Game Suppose one paper clip is on 5 – what products can you make by moving the other paper clip? List five multiples of 5 that are not on the game board. Suppose one paper clip is on 3 – what products can you make by moving the other paper clip? Circle the factors of the number 450 using your divisibility rules: 2 3 4 5 6 9 10 1.

Use product and factor in a sentence to describe P. O. D. If one of the paper clips is on 5, what products can you make by moving the other paper clip? Use product and factor in a sentence to describe 6 x 3 = 18. 1 2 3 4 5 6 7 8 9 10 12 14 15 16 18 20 21 24 25 27 28 30 32 35 36 40 42 45 48 49 54 56 63 64 72 81 Factors: 1 2 3 4 5 6 7 8 9

P. O. D. List 5 multiples of 5 that are not listed on the product game board. Use product and factor in a sentence to describe 9 x 6 = 54. 1 2 3 4 5 6 7 8 9 10 12 14 15 16 18 20 21 24 25 27 28 30 32 35 36 40 42 45 48 49 54 56 63 64 72 81 Factors: 1 2 3 4 5 6 7 8 9

Making your own Product Game Decide on a factor list. Figure out all products. Figure out board size and layout. Do a rough draft of your game. Do a neat final copy. Final copy due tomorrow.

The Factor Game 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 http://illuminations.nctm.org/ActivityDetail.aspx?ID=12

The best first move in the factor game is… P. O. D. The Factor Game The best first move in the factor game is… because…. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  

WARM UP 9-9-10 Your partner’s first move of the game is 28. 1. What factors do you get to circle? 2. What is your score? 3. What is their score? The Factor Game 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Using your divisibility rules, circle the factors of the number. 4.) 82 2 3 4 5 6 9 10 5.) 960 2 3 4 5 6 9 10

List the factor pairs for each number: 40. _______________________ 45 List the factor pairs for each number: 40 _______________________ 45 _______________________ 47 _______________________ Draw rectangles to represent the factor pairs of 32. Draw rectangles to represent the factor pairs of 17.

What is the best first move on the 49 board? P. O. D. What is the best first move on the 49 board? The 49 Board Factor Game 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49  

P. O. D. Using the words factor, multiple, product, and divisible by, write at least 4 facts about the number 48.  

The Smiths want to set up a tailgate area near the stadium The Smiths want to set up a tailgate area near the stadium. They want to use 8 square yards of space. What size rectangles can the Smiths use to set up their tailgate? 1 X 8 4 X 2 2 x 4 8 X 1

The Johnsons want their tailgate to be better than the Smiths The Johnsons want their tailgate to be better than the Smiths. Their tailgate will be 16 square yards of space. What size rectangles can the Johnsons use to set up their tailgate? 1 X 16 2 X 8 4 X 4 8 X 2 16 X 1

The Lewis family knows their tailgate is the best The Lewis family knows their tailgate is the best. They have TVs set up, grills, and tons of food. Their tailgate will be 30 square yards of space. What size rectangles can the Lewis family use to set up their tailgate? 5 x 6 2 x 15 6 x 5 15 x 2 10 x 3 30 x 1 3 x 10 1 x 30

Finding Patterns Among Factor Pairs Investigation 3.2 Finding Patterns Among Factor Pairs Find all factor pair rectangles for your number. Cut each out neatly from the grid paper. Paste to the number sheet. Label with the dimensions. Ex: 8 x 4 4 x 8

32: 36: 35: 40: 45: List the factor pairs for these numbers: Discussion Questions: What numbers between 1 and 30 are prime? Can you locate perfect squares just by looking at the factor rectangles? Which numbers have the most factors? Which pairs of primes differ by exactly 2? These are called _____________. Do larger numbers always have more factors than smaller numbers? 32: 36: 35: 40: 45:

Investigation 3.2 Follow-Up Questions What patterns do you see? Do you notice anything about even or odd numbers? Which numbers have the most rectangles? What kind of numbers are these? Which numbers have the fewest rectangles? What kind of numbers are these? Why do some numbers make squares? How do the dimensions of the rectangles relate to the number’s factors?

What size rectangles can be made for 32? 32 x 1 What size rectangles can be made for 32? 16 x 2 2 x 16 8 x 4 4 x 8 1 x 32

An amusement park wants to build a bumper car track that takes up 12 square meters of floor space. Tiles for the floor come in square meter shapes. Rails to surround the floor are each one meter long. With a neighbor, discuss and sketch some possible rectangular designs for this floor plan. For each design, label how many total floor tiles and how many rail sections you will need.

A different amusement park wants to build a bumper car track that takes up 18 square meters of floor space. With a neighbor, move your tiles, sketching all possible rectangular designs for this floor plan as you find them. For each design, label how many total floor tiles and how many rail sections you will need. Are there more possibilities if you do not keep with the rectangular shape? Find and sketch two more possibilities, labeling number of floor tiles and rail sections.

Looking at your plans for 18 square meters of space… Which of the rectangular designs do you think is the best shape for a bumper car area? Which of the nonrectangular designs would be the best for a bumper car area? Which of the rectangular designs requires the most rail sections? Which requires the least? Do they require the same amount of floor tiles? How would you persuade you’re a customer to buy your favorite design if you worked the design company?

HOMEWORK Sketch all rectangular options for a bumper car track with an area of 24 square meters. Draw at least two nonrectangular options that you think would be good designs. BONUS for a SPOT: Write a persuasive paragraph selling your favorite of these designs to an amusement park. Why should they pick your design?

Using your divisibility rules, circle each factor of the number Using your divisibility rules, circle each factor of the number. X out any numbers that are not factors. 252 2 3 4 5 6 9 10 648 2 3 4 5 6 9 10 870 2 3 4 5 6 9 10

Prime Factorization Notes Step 1: Using your divisibility rules, think about what prime numbers will go into the number. Step 2: Continue dividing the number by prime numbers until all you have left is prime numbers. Step 3: Write your answer using exponents. **You cannot use the number one for this because one is NOT prime!!! **Remember the prime numbers 2, 3, 5, 7, 11, 13*

One method: Making a Factor Tree Put the number at the top. Break the number down into a prime number times another factor and branch out. Circle all prime numbers as you go! Continue breaking it down until all you have left is prime numbers. Use exponents if you can to write the prime number factors. Ex. 100

Another Method: Using a Division Ladder Put the number at the top. Draw an upside-down division sign underneath it and divide it by a prime number. Write the new number underneath. All prime numbers should be down the left side. Continue breaking it down until all you have left is a prime number at the bottom. The numbers on the outside of the ladders are the prime factorization. Ex. 100

EXAMPLE of both methods on the same number: Factor Tree Division Ladder 200 200

YOU TRY SOME PRIME FACTORIZATION: Factor Tree Division Ladder 90 90 120 120

EXPONENTS An exponent is used to tell you how many times a base number is to be listed and used as a factor. Ex. 34 = 3x3x3x3 = ___ Ex. 53 = 5x5x5 = ___ Try this one: 26 = ____________ = _____ Exponents can be used to express prime factorization in a more concise manner. Ex. 2x2x3x5x5 would be written 22x3x52

Warm UP 10-13-08 Find the Prime Factorization of the Following using either the division ladder or a factor tree. Write your answer using exponents. 240 80 75

Greatest Common Factor and Least Common Multiple The GCF is simply the largest factor that two (or more) numbers share! The LCM is the smallest multiple that two or more numbers share! There are a variety of ways to find GCF and LCM, but in 6th grade we use prime factorization and a Venn diagram.

REAL LIFE PROBLEM SOLVING USING GCF AND LCM You have 27 Reese’s Cups and 66 M & M’s. Including yourself, what is the greatest number of friends you can enjoy your candy with so that everyone gets the same amount?

REAL LIFE PROBLEM SOLVING USING GCF AND LCM Miriam’s uncle donated 100 cans of juice and 20 packs of cheese crackers for the school picnic. Each student is to receive the same number of cans of juice and the same number of packs of crackers. What is the largest number of students that can come to the picnic and share the food equally? How many cans of juice and how many packs of crackers will each student receive?

REAL LIFE PROBLEM SOLVING USING GCF AND LCM Mrs. Armstrong and 23 of her students are planning to eat hot dogs at the upcoming DMS picnic. Hot dogs come in packages of 12 and buns come in packages of 8. What is the smallest number of packs of dogs and the smallest number of packs of buns Mrs. Armstrong can buy so that everyone INCLUDING HER can have the same number of hot dogs and there are no leftovers? How many dogs and buns does each person get?

Warm Up 1. You and a friend are shopping for new shirts Warm Up 1. You and a friend are shopping for new shirts. You find one you like that costs $5. Your friend finds one they like that costs $7. How many shirts would you each have to buy to spend the same amount of money? 2. You have 27 Reese’s Cups and 66 M&M’s. Including yourself, what is the greatest number of friends you can enjoy your candy with so that everyone gets the same amount? 3. Miriam’s uncle donated 100 cans of juice and 20 packs of cheese crackers for the school picnic. Each student is to receive the same number of cans of juice and the same number of packs of crackers. What is the largest number of students that can come to the picnic and share the food equally? How many cans of juice and how many packs of crackers will each student receive?

Fun Problem – Warm Up Write down the number of the month in which you were born. Multiply that number by 4. Add 13. Multiply by 25. Subtract 200. Add the day of the month on which you were born. Multiply by 2. Subtract 40 Multiply by 50 Add the last two digits of the year in which you were born. Subtract 10,500. Does the number look familiar????

Warm Up 9-25-09 1. How do you determine if a number is prime? 2. Using divisibility rules, circle all factors of these numbers: 456 2 3 4 5 6 9 10 1332 2 3 4 5 6 9 10 3. What is the prime factorization (using exponents) of the number 150?

PPOP QUIZ Divis Rules 9-8-10 Using divisibility rules, circle all factors of these numbers: 1.) 456 2 3 4 5 6 9 10 2.) 1332 2 3 4 5 6 9 10 3.) 585 2 3 4 5 6 9 10 Write the divisibility rule for 3 = ____________________________ 6 = ____________________________ 4 = ____________________________

POP QUIZ What is the prime factorization of 240? Use exponents to write your answer. 2. Use prime factorization to solve: What is the GCF of 40 and 60? What is the LCM of 6 and 9? Circle the factors of the number. Cross out numbers that are not factors. 558 2 3 4 5 6 9 10 4. Is the number 28 prime or composite? Why or why not? Explain your answer in complete sentences.

POP QUIZ What is the prime factorization of 350? Use exponents to write your answer. 2. Use prime factorization to solve: What is the GCF of 20 and 50? What is the LCM of 7 and 8? Circle the factors of the number. Cross out numbers that are not factors. 1008 2 3 4 5 6 9 10 4. Is the number 498 prime or composite? Why or why not? Explain your answer in complete sentences.

Warm Up 10---2008 Linus always waits in the pumpkin patch for the Great Pumpkin to arrive on Halloween. The great pumpkin came by early this year and hid toys for Linus. Can you find which pumpkin he hid toys in for Linus? He didn’t hide it in the fourth pumpkin from either end. He didn’t hide it in the pumpkin to the left of center. The pumpkin he hid them in had at least three pumpkins on either side. The pumpkin he hid it in was not next to or on the end of the vine.

Warm Up 10----2008 How far can a bat travel in 7 hours if it is flying at twenty-nine miles per hour? Jonathan gave away one hundred twenty six pieces of candy on Halloween. He gave six pieces to each child. How many children visited his house? Robert bought 4 big bags of candy and each bag had 58 pieces in it. On Halloween, 24 children came to Robert’s house and he gave them each 3 pieces of candy. How much does he have left?

Using divisibility rules, circle all factors of these numbers: 456 2 3 4 5 6 9 10 1353 2 3 4 5 6 9 10 2. What is the prime factorization (using exponents) of the number 500? What is the GCF and LCM of the numbers 12 and 30?