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					                                             Unit 1
                                          Prime Time
                                            OBJECTIVES

•Understand relationships among factors, multiples, divisors and products
•Recognize and use properties of prime and composite numbers, even and odd numbers, and square
numbers
•Use rectangles to represent the factor pairs of numbers
•Develop strategies for finding factors and multiples, least common multiples, and greatest common
factors
•Recognize and use the fact that every whole number can be written in exactly one way as a product
of prime numbers
•Use factors and multiples to solve problems and to explain some numerical facts of everyday life
•Develop a variety of strategies for solving problems-building models, making lists and tables, drawing
diagrams, and solving simpler problems


 Mrs. Carol Lange                                                                                         1
                                  Unit 1
                               Prime Time
                                   Big Ideas
Factors and Products
Whole Number Patterns and Relationships
Common Multiples and Factors
Factorizations: Searching for Factor Strings

                             Essential Questions
Will breaking a number into factors help me solve a problem?
What relationships are revealed with factors?
What do the factor and multiples of the numbers tell me about the situations?
How can I find the factors of the numbers?
How can I find the multiples?
What common factors and common multiples do the numbers have?

Mrs. Carol Lange                                                                2
 Textbook page 4
                                      Unit 1
                                   Prime Time
                                          UNIT PROJECT
Choose a whole number that you especially like:
•Your number must be between 10 and 100
•Record your number
•Explain why you chose that number
•List three or four mathematical fact about your number
•List three or four connections you can make between your number and your world
•Add to your notes as you progress through the unit
•At the end of the unit you will be creating a project to highlight your number



Mrs. Carol Lange                                                                  3
 Textbook page 5, 70
                                   Unit 1
                                Prime Time
                       ESSENTIAL VOCABULARY
             factor                common multiples
             divisor               least common multiple
             multiple              common factors
             prime number          greatest common factor
             composite number      factorizations
             factor pair           prime factorizations
             square numbers        exponent
             even number           relatively prime
             odd number            conjecture
             Venn diagram



Mrs. Carol Lange                                            4
 Textbook page 74-79
                                   Unit 1
                                Prime Time
                          INVESTIGATION ONE
                             Factors and Products
•Factors are whole number divisors of numbers:
          1,2,5,10 are factors of 10
          Every number has a limited number of factors!
•Play The Factor Game to practice finding factors, and to see
          relationships among numbers
          http://www.PHSchool.com          webcode amd-1101
•Proper factors are all the factor of a number except the number itself:
          1,2,5 are the proper factors of 10
•Multiples are the products of a whole number and another whole number: the
          multiples of 5 are 5 (5x1), 10 (5x2), 15 (5x3), and so on. Every
          number has an unlimited number of multiples.
•Play The Product Game to practice finding multiples, and to see
          relationships among numbers
          http://www.PHSchool.com          webcode amd-1103

Mrs. Carol Lange                                                              5
 Textbook page 6-21
                                   Unit 1
                                Prime Time
                          INVESTIGATION ONE
                            Factors and Products
•Prime Numbers are numbers that have exactly two factors:
         3,5,7,11,13,17,19…are prime because the only divisors they have are
         1 and the number itself
         http://www.PHSchool.com       webcode ame-9031
•The number one is neither prime, nor composite
•Composite Numbers are numbers that have more than two factors:
         2,4,6,8,9,10,12,14,15…are composite because they have more than
         two factors (divisors)




Mrs. Carol Lange                                                               6
 Textbook page 6-21
                                  Unit 1
                               Prime Time                                  Always
                                                                          support
                                                                            your
                                INVESTIGATION ONE                         answers
                                                                            with
                                                                         examples!
                       Sample ACE questions and answers:
APPLICATIONS:

5) What factor is paired with 6 to give it 54?
          9, because 9 x 6 = 54

16) Why is the set of factors of a number not the same as the set of proper
factors of that number?
          Factors are different than proper factors because they do not include
          the number itself.
          Factors of 9 are 1,3,and 9; proper factors of 9 are 1,3
                               Multiple-choice skills practice:
                       http://www.phschool.com webcode ama-1154



Mrs. Carol Lange                                                                     7
 Textbook page 14-15
                               Unit 1
                            Prime Time                                      Always
                                                                           support
                                                                             your
                             INVESTIGATION ONE                             answers
                                                                             with
                                                                          examples!
                    Sample ACE questions and answers:
CONNECTIONS:

33A) In developing the ways in which we calculate time,
astronomers divided an hour into 60 minutes. Why is 60 a good choice
(better than 59 or 61)?
          60 is a good choice for calculating time, because it has many factors
          (1,60; 2,30; 3,20; 4,15; 5,30; 6,10). 59 and 61 do not have as many
          factors.
33B) If you could select another number to represent the number of
minutes in an hour, what would be a good choice? Why?
          I’d pick a number like 100, because it has a lot of factors (1,100;
          2, 50; 4,25; 5,20; 10,10)


Mrs. Carol Lange                                                                      8
 Textbook page 18
                              Unit 1
                           Prime Time                                       Always
                                                                           support
                                                                             your
                             INVESTIGATION ONE                             answers
                                                                             with
                                                                          examples!
                    Sample ACE questions and answers:
EXTENSIONS:


43) What is the best move (in the Factor Game) on a 49-board? Why?

          47 is a good first move, because it is the greatest prime number. You
          will get 47 points, and your opponent will get only 1.




Mrs. Carol Lange                                                                      9
 Textbook page 19
                              Unit 1
                           Prime Time
                         INVESTIGATION TWO
                 Whole-Number Patterns and Relationships

•Factor Pairs are two whole numbers that are multiplied to get a product
         factor pairs of 10 are 1,10 and 2,5
         Remember: every number has a limited number of factors!
•Square Numbers are numbers that result from multiplying the same number
         times itself. (4 x 4 = 16, 2 x 2 = 4)
•Odd Numbers are numbers that do not have two as a factor.
•Even Numbers are numbers that have two as a factor.
•A conjecture is a best guess or prediction based on an observed pattern
•A Venn diagram uses circles to group things that belong together




Mrs. Carol Lange                                                           10
 Textbook page 22-36
                                  Unit 1
                               Prime Time                                  Always
                                                                          support
                                                                            your
                                  INVESTIGATION TWO                       answers
                                                                            with
                       Sample ACE questions and answers:                 examples!
APPLICATIONS:

8) What type of number has an odd number of factors? Give
examples
         Square numbers have an odd number of factors, because one factor
         pair is always the same. (16: 1,2, 4, 8, 16)

15) How can you determine whether a sum of several numbers, such as
13 + 45 + 24 + 17 is even or odd?
          If all the numbers in the ones place are even, or if there is an even
          number of odd numbers, than the sum is even.
          For example: The sum of 3 + 5 + 4 + 7 will be odd, because there are
          three odd addends.
            Problem 15 help:
   http://www.phschool.com webcode ama-1215

Mrs. Carol Lange                                                                     11
 Textbook page 31
                                Unit 1
                             Prime Time                                        Always
                                                                              support
                                                                                your
                              INVESTIGATION TWO                               answers
                                                                                with
                                                                             examples!
                     Sample ACE questions and answers:
CONNECTIONS:

28) Allie’s eccentric aunt, May Belle, hides $10,000 in $20 bills
under her mattress. If she spends one $20 bill every day, how many days
will it take for her to run out of bills?
           It will take 500 days. I made a chart for 5 days, and saw that it would
           take 5 days to spend $100. 100 goes into 10,000, 100 times, so 100
           times 5 days equals 500 days.
           Be sure to explain any strategy you use!




Mrs. Carol Lange                                                                         12
 Textbook page 33
                              Unit 1
                           Prime Time                                     Always
                                                                         support
                                                                           your
                            INVESTIGATION TWO                            answers
                                                                           with
                                                                        examples!
                    Sample ACE questions and answers:
EXTENSIONS:

37) For any three consecutive numbers, what can you say about
odd numbers and even numbers? Why?

          For any three consecutive numbers, either two of the numbers are
          odd (4,5,6) or two of the numbers are even (5,6,7). It depends on
          what the first number is.




Mrs. Carol Lange                                                                    13
 Textbook page 34
                                  Unit 1
                               Prime Time
                               ORDER OF OPERATIONS
        Order of Operations is a rule developed by mathematicians that
                 determines in what order you solve an equation.

  1) Solve all operations in parentheses first.
  2) Solve all exponents
  3) Multiply/divide left to right
  4) Add/subtract left to right
               Do ALL multiplication before you add or subtract!
Solve these:
7+6–3x2=              7   (multiply 3 x 2 first, then add 7 + 6 , then subtract 6 from 13)

32 – 2 x 3 + 6=       9   (Solve 32 first to get 9, then multiply 2 x 3 to get 6; 9 – 6 + 6 = 9)




  Mrs. Carol Lange                                                                                 14
   Textbook page 26
                               Unit 1
                            Prime Time
                          INVESTIGATION THREE
                  Common Multiples and Common Factors

•Common Multiples are multiples that appear on the list of multiples for two (or
       more) different numbers. Some multiples of 5 are 5, 10, 15, 20, and
       some multiples of 10 are 10, 20, 30. Common multiples for 5 and 10
       are 10, and 20 because they are on both lists. Numbers have an
       infinite number of common multiples. The smallest number on both
       lists is the Least Common Multiple.
•Common Factors are factors that appear on the list of factors for two (or
       more) different numbers. Factors of 18 are 1,18; 2,9; 3,6 and the
       factors of 12 are 1,12; 2, 6; 3,4 so common factors are 1, 2, 3, and 6.
       the Greatest Common Factor is 6.



Mrs. Carol Lange                                                                   15
 Textbook page 37-48
                                Unit 1
                             Prime Time                                  Always
                                                                        support
                                                                          your
                            INVESTIGATION THREE                         answers
                                                                          with
                      Sample ACE questions and answers:
                                                                       examples!
APPLICATIONS:

6) List the common multiples from 1 to 100 for each pair of
numbers. Then find the least common multiple for each pair.

          20 and 25

          20: 20, 40, 60, 80, 100
          25: 25, 50, 75, 100
          The LCM is 100.
          It is the smallest multiple on that appears on both lists.




Mrs. Carol Lange                                                                   16
 Textbook page 42
                                     Unit 1
                                  Prime Time                 Always
                                                            support
                                                              your
                                 INVESTIGATION THREE        answers
                                                              with
                       Sample ACE questions and answers:   examples!
CONNECTIONS:

36) 3 x 5 x 7 = 105 Use this fact to find each product.
          a) 9 x 5 x 7 =
          105 x 3 ( 315) because 9 is 3 x 3.
          You need to include one more factor of 3.
          b) 3 x 5 x 14 =
          105 x 2 (210) because 14 is 7 x 2.
          You need to include one more factor of 2.
          c) 3 x 50 x 7 =
          105 x 10 (1050) because 50 is 10 x 5.
          You need to include one more factor of 10.
          d) 3 x 25 x 7 =
          105 x 5 (525) because 25 is 5 x 5.
          You need to include one more factor of 5.
Mrs. Carol Lange                                                       17
 Textbook page 45
                             Unit 1
                          Prime Time                                        Always
                                                                           support
                                                                             your
                           INVESTIGATION THREE                             answers
                                                                             with
                    Sample ACE questions and answers:                     examples!
EXTENSIONS:

38) Ms. Santiago has many pens in her desk drawer. She says that
    if you divide the total number of pens by 2, 3, 4, 5, or 6, you get a
    remainder of 1. What is the smallest number of pens that could be in
    Ms. Santiago’s desk?

          The smallest number of pens in Ms. Santiago’s desk is 61. You need
          to find the least common multiple of 2, 3, 4, 5 and 6, and add 1 (the
          remainder) to it.




Mrs. Carol Lange                                                                      18
 Textbook page 45
                               Unit 1
                            Prime Time
                          INVESTIGATION FOUR
                 Factorizations: Searching for Factor Strings

•Factorizations are strings of factors of a number. All numbers can be written
          as strings of factors. For example: 50 can be written as 2 x 25, or
          2 x 5 x 5.
•All numbers are products of prime numbers. The strong of factors that is made
          up of all prime numbers is called the Prime Factorization of a
          number. Prime Factorizations can be found by dividing each factor by
          a prime number, or by using a factor tree to identify factor pairs.
•Exponents tell you how many times a number is used as a factor.
          For example, 2 x 2 x 2 is written in exponent form as 23.
•Number pairs whose greatest common factor is one, are called relatively
          prime. Relatively prime numbers can be multiplied together
          to find their least common multiple.
Mrs. Carol Lange                                                                 19
 Textbook page 49-60
                               Unit 1
                            Prime Time                                    Always
                                                                         support
                                                                           your
                             INVESTIGATION FOUR                          answers
                                                                           with
                    Sample ACE questions and answers:                   examples!
APPLICATIONS:

13) To indicate multiplication, you can use a raised dot symbol.
For example, 3 x 5 = 3 . 5. Find the prime factorization of 312 using
raised dot symbols.
                                        312

                                       6         52

                                   3       2    2     26

                                                  2    13
                    3 . 2 . 2 . 2 . 13 = 312   or 3 . 23 . 13 = 312

Mrs. Carol Lange                                                                    20
 Textbook page 56
                                Unit 1
                             Prime Time                                          Always
                                                                                support
                                                                                  your
                              INVESTIGATION FOUR                                answers
                                                                                  with
                     Sample ACE questions and answers:                         examples!
CONNECTIONS:

32) What is my number?
a) My number is a multiple of 2 and 7.
         Start by listing multiples of 2 and 7: 14, 28, 42, 56, 70, 84, 98
b) My number is less than 100 but greater than 50.
         Using the list from question a, eliminate multiples less than 50:
         56, 70, 84, 98
c) My number is the product of 3 different prime numbers.
         Find the prime factorization of each of your multiples.
         56 (2 x 2 x 2 x 7); 70 (2 x 5 x 7); 84 (2 x 2 x 3 x 7); 98 (2 x 7 x 7).
         Only 70 is a product of three different prime numbers.


Mrs. Carol Lange                                                                           21
 Textbook page 58
                                  Unit 1
                               Prime Time
                                  INVESTIGATION FIVE
                              THE LOCKER PROBLEM
Students in a school create a problem by opening 1000 lockers.
•student 1 runs down the row of lockers and opens every door.
•student 2 closes the doors of all the even number lockers.
•student 3 changes the state (opens or closes) the doors of the lockers of
            multiples of 3.
•Student 4 changes the state of the doors of lockers that are multiples of 4.

•Student 5 changes the state of every 5th door.
   When all the students have finished, which locker doors are opened?
                    Discuss strategies you can use to solve the problem.

                            Make a conjecture about the answers!

NEED HELP? http://www.PHSchool.com                       webcode amd-1501


Mrs. Carol Lange                                                                22
 Textbook page 61-64
                              Unit 1
                           Prime Time
                                                                      Always be
                                                                        able to
                          END OF UNIT REVIEW                         provide an
                                                                     explanation
                                                                       for your
                        For your Unit Test, know how to:                work!



•Find factors and factor pairs of a number
•Find common factors and the greatest common factor of two numbers
•Find multiples of a number and least common multiples of two or more numbers
•Use exponents and Order of Operations
•Find a factor string and prime factorization of a number

            Use your study guide from the back of the parent letter!




Mrs. Carol Lange                                                                   23
 Textbook page 6-60
                                Unit 1
                             Prime Time
                             SPECIAL NUMBER PROJECT
For your special number project you need to choose one of the following projects:
          •Collage
          •Comic strip
          •Narrative story
               In your project, you need to include as much information
                         about your special number as possible.
                   You must include the following vocabulary words:
factor               multiple            prime               composite
factor pair          common factor       common multiple     even number
odd number           prime factorization square number       Venn diagram

Mrs. Carol Lange                                                                    24

				
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