Outline

Introduction

Through a series of problems faced by students at Avatar Middle we will learn how to use factors, multiples, arrays, prime and composite numbers as well as exponents.

Student Activities Activities the students will do to help them remember how to use factors, multiples... this will be the time that the students may ask questions. Some activities will be done as a group.

Activity 1: Back to School!

Part 1: Music You and your friends have tickets to attend a music concert. While standing in line, the promotion states he will give a free album download to each person that is a multiple of 2. He will also give a backstage pass to each fourth person and floor seats to each fifth person. Which person will receive the free album download, backstage pass, and floor seats? Explain the process you used to determine your answer.

Part 2: School Supplies The Parents Teachers Association (PTA) at your school donated school supplies to help increase student creativity and student success in the classroom. Your teacher would like you to create kits that include one package of colored pencils, one glue stick, and one ruler. When you receive the supplies, you notice the colored pencils are packaged 12 boxes to a case, the rulers are packaged 30 to a box, and glue sticks are packaged 4 to a box.

1. What is the smallest number of each supply you will need in order to make the kits and not have supplies left over? Explain your thought process. 2. How many packaged rulers, colored pencils, and glue sticks will you need in order to make the kits? Explain the process you used to determine how many packages are needed for each supply.

Activity 2:

Part I: Arrays, Factors, and Number Theory

Create, draw or shade all possible arrays for the numbers 1-20.Label all of the dimensions of the arrays, which are the factors of each number. Look for patterns in the arrangements, factors, or drawings. Describe the patterns or observations that help you "see" the factors, prime numbers, composite numbers and square numbers. In the numbers 1-20, label the prime, composite, and square numbers. Describe all the things you notice about the arrays and patterns, but especially discuss what you notice about the number 1.
 * Part II: **

1. Using your previous work from **Arrays, Factors and Number Theory, **list all the factors in order from least to greatest, for each number 1-20. 2. Choose any two numbers from your list of 1-20. What factors are in both lists? 3. What is the largest factor that they have in common? 4. Try this on several other pairs of numbers from 1-20. 5. Can you do it for 3 of the numbers? Try it for 3 numbers. 6. When would it be useful to know the common factors or the greatest common factor of two or more numbers? 7. What advice would you offer to a friend who was having trouble finding all the factors of any number?

Collaborative Project for Assessment You will have to use your knowledge of number theory to get past three challenges. The first challenge will be to tell what the Fundamental Theorem of Arithmetic is. The second challeng will be to find the Secret Number. The final challenge will be Slammin Lockers.

Part I: "Hi Mike! Let me tell you about the Fundamental Theorem of Arithmetic!"

The fundamental theorem of arithmetic states that every natural number greater than one is either prime or can be written as a unique product of prime factors. What does this mean? Refer to the work you did in previous problems to help you explain the fundamental theorem of arithmetic to your friend, Mile, who has been absent. Be sure to include the following terms: factor, multiple, divisible, prime, composite, prime factorization and exponents.

Part II: Secret Number

Juanita has a secret number. Read her clues and then answer the questions that follow: Juanita says, "Clue 1" My secret number is a factor of 60." 1. Can you tell what Juanita‟s secret number is? Explain your reasoning. 2. Daren said that Juanita‟s number must also be a factor of 120. Do you agree or disagree with Daren? Explain your reasoning. 3. Malcolm says that Juanita‟s number must also be a factor of 15. Do you agree or disagree with Malcolm? Explain your reasoning. 4. What is the smallest Juanita‟s number could be? Explain. 5. What is the largest Juanita‟s number could be. Explain. 6. Suppose for Juanita‟s second clue she says, " Clue 2: My number is prime." 7. Can the class guess her number and be certain? Explain your answer. 8. Suppose for Juanita‟s third clue she says, "Clue 3: 15 is a multiple of my secret number." 9. Now can you tell what her number is? Explain your reasoning. 10. Your secret number is 36. Write a series of interesting clues using factors, multiples, and other number properties needed for somebody else to identify your number.

Part III: Slammin‟ Lockers

Georgia Middle School has 100 students with lockers numbered 1 through 100. One day, Sally walks down the hall and opens all the lockers. Eric goes behind her and closes all the lockers with an even number. Then, Jane changes the situation of the lockers with numbers that are multiples of 3. This means that a closed locker is opened and an open locker is closed. If this pattern continues FOR ALL 100 STUDENTS, which lockers will remain open after the 100th student walks down the hall? Explain your thinking giving details, and using both appropriate mathematical models and language. What if there were 500 students and 500 lockers? What if there were 1000 students and 1000 lockers? Can you find a rule for any number of students and lockers? Explain why your rule works.