Patterns and Inductive Reasoning - PowerPoint - PowerPoint by sJq1Gt5I

VIEWS: 97 PAGES: 20

									      Patterns and Inductive Reasoning
GEOMETRY LESSON 1-1

                                                    (For help, go the Skills Handbook, page 715.)


Here is a list of the counting numbers: 1, 2, 3, 4, 5, . . .
Some are even and some are odd.
    1. Make a list of the positive even numbers.

    2. Make a list of the positive odd numbers.

    3. Copy and extend this list to show the first 10 perfect squares.
             12 = 1, 22 = 4, 32 = 9, 42 = 16, . . .

    4. Which do you think describes the square of any odd number?
                  It is odd.            It is even.


         Make sure your name is in your book!



                                            1-1
            Patterns and Inductive
                  Reasoning
GEOMETRY LESSON 1-1




Solutions

1. Even numbers end in 0, 2, 4, 6, or 8: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, . . .

2. Odd numbers end in 1, 3, 5, 7, or 9: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, . . .

3. 12 = (1)(1) = 1; 22 = (2)(2) = 4; 32 = (3)(3) = 9; 42 = (4)(4) = 16;
   52 = (5)(5) = 25; 62 = (6)(6) = 36; 72 = (7)(7) = 49; 82 = (8)(8) = 64;
   92 = (9)(9) = 81; 102 = (10)(10) = 100

4. The odd squares in Exercise 3 are all odd, so the square of any odd
   number is odd.




                                            1-1
               What is Geometry
•Geometry is more than the study of shapes. It is the study of
truths.
•These truths are constant, no matter what the situation.
•Geometry uses reason (logic) to prove truths and to build
upon them to prove even more truths.
•The study of geometry is a study of how to think logically.

 There are two types of logical strategies:
 1. Inductive Reasoning
 2. Deductive Reasoning
     Chapter 1 Section 1

• Goals:
• Use inductive reasoning to
  make a conjecture.
               Vocabulary 1.1

•Inductive Reasoning •investigating using the observation of
                     patterns
•Conjecture          •A conclusion reached based upon
                     inductive observation
•Counterexample      •An example that shows the conjecture is
                     not correct
•Prime Number        •A Positive number with no factors other
                     than itself and 1.
                     (The smallest prime number is 2.)
       Use Inductive Reasoning:
GEOMETRY LESSON 1-1




                      Find a pattern for the sequence. Use the pattern to
             show the next two terms in the sequence.
         384, 192, 96, 48, …

    #                              1. Write the sequence
    384               ÷2           2. What value is +,-,x, or ÷ each
    192               ÷2              time?
    96                ÷2
    48                ÷2
     24               ÷2
       12
Each term is half the preceding term. So the next two terms are
48 ÷ 2 = 24 and 24 ÷ 2 = 12.             1-1
      Use Inductive Reasoning
GEOMETRY LESSON 1-1




                        Make a conjecture about the sum of the cubes of the first 25
               counting numbers.
  Find the first few sums.


  13                            =1      = 12      = (1)2
  13 + 23                       =9      = 32      = (1+2)2
  13 + 23 + 33                  = 36    = 62      = (1+2+3)2
  13 + 23 + 33 + 43             = 100   = 102     = (1+2+3+4)2
  13 + 23 + 33 + 43 + 53        = 225   = 152     = (1+2+3+4+5)2

  The sum of the cubes equals the square of the sum of the
  counting numbers.




                                            1-1
        Use Inductive Reasoning
GEOMETRY LESSON 1-1




                       The first three odd prime numbers are 3, 5, and 7. Make and
              test a conjecture about the fourth odd prime number.


  One pattern of the sequence is that each term equals the preceding term plus 2.

  So a possible conjecture is that the fourth prime number is 7 + 2 = 9.

  However, because 3 X 3 = 9 and 9 is not a prime number, this conjecture is false.

  By applying the assumed pattern and then testing the result against the initial
  directions, we have found a counterexample.
  A counterexample applies the presumed pattern and gives a false result.
  Only ONE counterexample is needed to prove a conjecture is false.
  Conjecture: odd prime numbers are found by adding 2 to each odd prime.
  Counterexample: 7 is odd prime, 7+2 = 9, 9 is not prime.
  Result: Conjecture is false.
  The fourth prime number is 11.
                                          1-1
 When points on a circle are joined, they produce unique
 regions within the circle:

Points    Regions
   2        2
   3         4
   4         8
   5        16
   6        ??

Will the # of regions always be twice as many as the
previous number?

          6 points yields 30 regions.
          30 is NOT 2x16!
          Conjecture is false.
        Use Inductive Reasoning
GEOMETRY LESSON 1-1




                       The price of overnight shipping was $8.00 in 2000, $9.50 in
               2001, and $11.00 in 2002. Make a conjecture about the price in 2003.

 year              $$
                                            Write the data in a table. Find a pattern.
2000            8.00          + 1.50
                                            Each year the price increased by $1.50.
2001            9.50          + 1.50
                                           A possible conjecture is that the
2002            11.00         + 1.50       price in 2003 will increase by
                                           $1.50.
2003             12.50
                                           If so, the price in 2003 would be
                                           $11.00 + $1.50 = $12.50.
                     Re-Cap
•Inductive Reasoning •Based upon observation of patterns
•Conjecture          •A conclusion reached based upon
                     inductive observation
•Counterexample      •An example that shows the conjecture is
                     not correct
•Prime Number        •Number with no factors other than itself
                     and 1.



Tips for Inductive Reasoning:
•Make a list
•Make a table when comparing two sets of numbers
•Look for simple numbers patterns
Additional Practice
GEOMETRY LESSON 1-1




Find a pattern for each sequence.   Use the table and inductive reasoning.
Use the pattern to show the next
two terms or figures.

    1. 3, –6, 18, –72, 360

         –2160; 15,120              3. Find the sum of the first 10 counting numbers.
                                          55
                                    4. Find the sum of the first 1000
    2.                                 counting numbers.
                                          500,500
                                    Show that the conjecture is false by finding one
                                    counterexample.
                                    5. The sum of two prime numbers is an
                                       even number.
                                          Sample: 2+3=5, and 5 is not even


                                        1-1
                   Homework 1.1
        Homework, due at the beginning of the NEXT class:
        page 6
                                      Name
                                      Section #
                                      Page #
Remember,
this is an
Honor Code
                         Show your work                Make sure your
School!
                         here IN PENCIL                name is in your
                                                       book!
No Copying!

                 I pledge that I have neither given
                 nor received aid on this assignment
Check in ink!
GEOMETRY LESSON 1-1




Pages 6–9 Exercises
   1. 80, 160           12. 1 , 1               19. The sum of the first 6 pos.
                              5 6
                                                    even numbers is
   2. 33,333; 333,333   13. James, John
                                                    6 • 7, or 42.
   3. –3, 4             14. Elizabeth, Louisa
   4.   1, 1            15. Andrew, Ulysses     20. The sum of the first 30 pos.
        16 32
                                                    even numbers is
   5. 3, 0              16. Gemini, Cancer
                                                    30 • 31, or 930.
   6. 1, 1              17.
            3
   7. N, T                                      21. The sum of the first 100
                                                    pos. even numbers is
   8. J, J              18.
                                                    100 • 101, or 10,100.
   9. 720, 5040
   10. 64, 128
   11. 1 , 1
        36 49


                                     1-1
Check in ink!
GEOMETRY LESSON 1-1




22. The sum of the first          28. 1 ÷1 =3 and3 is        31. 31, 43
                                      2   3   2    2
    100 odd numbers is                improper.
    1002, or 10,000.                                         32. 10, 13
                                  29. 75°F
                                                             33. 0.0001, 0.00001
23. 555,555,555
                                  30. 40 push-ups;
24. 123,454,321                                              34. 201, 202
                                      answers may vary.
                                      Sample: Not very       35. 63, 127
25–28. Answers may vary.
       Samples are given.             confident, Dino may    36. 31 , 63
                                                                 32 64
                                      reach a limit to the
                                                             37. J, S
25. 8 + (–5 = 3) and 3 > 8
                       /              number of push-ups
26.     1 • 1 > 1 and 1 • 1 > 1       he can do in his       38. CA, CO
              /             / 2
        3 2 3         3 2
27. –6 – (–4) < –6 and                allotted time for
                                                             39. B, C
                                      exercises.
      –6 – (–4) < –4



                                          1-1
Check in ink!
GEOMETRY LESSON 1-1




40. Answers may vary.              42.          47. Answers may vary.
    Sample: In Exercise                             Samples are given.
    31, each number                                 a. Women may soon outrun
    increases by increasing                            men in running competitions.
    multiples of 2. In Exercise    43.              b. The conclusion was based
    33, to get the next term,
                                                       on continuing the trend
    divide by 10.
                                   44.                 shown in past records.
                                                    c. The conclusions are
41.                                                    based on fairly recent
                                                       records for women,
                                   45.                 and those rates of
                                                       improvement may not
      You would get a third line                       continue. The conclusion
      between and parallel to                          about the marathon is most
      the first two lines.         46. 102 cm          suspect because records
                                                       date only from 1955.


                                         1-1
Check in ink!
GEOMETRY LESSON 1-1




48. a.                         50. His conjecture is 52.   21, 34, 55
                                   probably false
                                   because most      53.   a. Leap years are years
                                   people’s growth            that are divisible by 4.
                                   slows by 18 until
                                   they stop growing       b. 2020, 2100, and 2400
                                   somewhere between
     b. about 12,000 radio         18 and 22 years.        c. Leap years are years
        stations in 2010                                      divisible by 4, except
     c. Answers may vary.      51. a.                         the final year of a
        Sample: Confident;                                    century which must
        the pattern has held                                  be divisible by 400.
        for several decades.                                  So, 2100 will not be a
                                                              leap year, but 2400
49. Answers may vary.                                         will be.
    Sample: 1, 3, 9, 27,           b. H and I
    81, . . .
    1, 3, 5, 7, 9, . . .           c. a circle

                                        1-1
Check in ink!
GEOMETRY LESSON 1-1




54. Answers may vary.                             55. (continued)
    Sample:                                           d.

     100 + 99 + 98 + … + 3 + 2 + 1
       1 + 2 + 3 + … + 98 + 99 + 100
     101 + 101 + 101 + … + 101 + 101 + 101
                                                  56. B
     The sum of the first 100 numbers is
                                                  57. I
      100 • 101 , or 5050.
          2
     The sum of the first n numbers is n(n+1) .   58. [2] a. 25, 36, 49
                                           2              b. n2
55. a. 1, 3, 6, 10, 15, 21                            [1] one part correct
    b. They are the same.
    c. The diagram shows the product of n
       and n + 1 divided by 2 when
       n = 3. The result is 6.


                                           1-1
Check in ink!
GEOMETRY LESSON 1-1




59. [4] a. The product of 11       59. (continued)             60-67.
           and a three-digit       [3] minor error in
           number that begins          explanation
           and ends in 1 is a
           four-digit number       [2] incorrect description
           that begins and ends        in part (a)
           in 1 and has middle
           digits that are each    [1] correct products for
           one greater than the        (151)(11), (161)(11),
           middle digit of the         and (181)(11)
           three-digit number.
           (151)(11) = 1661
           (161)(11) = 1771                                       68. B

         b. 1991                                                  69. N

         c. No; (191)(11) = 2101                                  70. G



                                           1-1
         Coordinate Assignments
Write your coordinate down. You will use this all year long! Put it on your
math folder!




      (1,-1)       (-1,2)        (-1,-3)      (1,4)        (1,0)

      (2,-1)       (-2,2)        (-2,-3)      (2,4)        (0,5)

      (3,-1)       (-3,2)        (-3,-3)      (3,4)        (-3,0)

      (4,-1)       (-4,2)        (-4,-3)      (4,4)        (0,-5)

								
To top