Strengthening Multiplicative Reasoning with Prime Numbers
Strengthening Multiplicative Reasoning with Prime Numbers Matt Roscoe University of Montana [email protected]
Ziv Feldman Boston University [email protected]
NCTM Research Conference San Francisco, CA April 15, 2016
1
Rationale
2
Setting the Stage
3
Explore the Landscape of Prime Factorizations 1-100
4
Constructing Factor Lists Using Primes
5
What It’s All About: The FTA
6
Conclusion
Rationale
“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” — G. Hardy (from A Mathematician’s Apology, London 1941).
Rationale
“...in mathematics, the primary subject-matter is not the mathematical objects but rather the structures in which they are arranged” (Resnik, 1999, p.201).
Rationale
Common Core State Standards for Mathematics (2010) Operations and Algebraic Thinking (4.OA) 4. Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite. (p. 29)
Rationale
Common Core State Standards for Mathematics (2010) Operations and Algebraic Thinking (4.OA) 4. Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite. (p. 29)
Rationale
A richer understanding of factors and multiples can be developed by drawing attention to the important role that prime numbers play when two numbers are multiplicatively compared.
Setting the Stage
Any whole number greater than zero with exactly two distinct, positive divisors is a prime number, or simply prime. Examples: 2, 3, 5, 7, 11
Setting the Stage
Any whole number greater than zero that has a positive factor other than 1 and itself is a composite number, or simply composite. Examples: 4, 6, 8, 10, 12
Setting the Stage The Sieve of Eratosthenes 1
Create a list of consecutive whole numbers starting from 2 to any n
2
The first prime is 2, let p “ 2
3
Circle p. Starting from p count up in multiples of p and eliminate each of these numbers (i.e. 2p, 3p, 4p, ....) until you reach n
4
Next, find the first number greater than p in the list that is not marked; let p equal this number and repeat the previous step
5
Continue this process until the list is exhausted, the circled numbers are the prime numbers up to n
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Setting the Stage
Factor Tree Method
Setting the Stage
Factor Lattice Method 72
Setting the Stage
Factor Lattice Method
2
72 36
Setting the Stage
Factor Lattice Method
2 2
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Setting the Stage
Factor Lattice Method
2 2 2
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Setting the Stage
Factor Lattice Method
2 2 2 3
72 36 18 9 3
Setting the Stage
Factor Lattice Method
2 2 2 3
72 36 18 9 3
Explore the Landscape of Prime Factorizations 1-100
You each are given a collection of numbers. Do the following: 1
2
Construct each number as a product of primes and find a collection of blocks to represent the number and construct a “prime tower”. Once you have constructed all your towers, discuss the following questions: Which pairs of numbers share factors? Which pairs do not? How do you know? 2 How can you find the largest factor that a pair of numbers share? How do you know? 3 Are there any pairs where one number is a multiple of the other? How do you know? 1
Explore the Landscape of Prime Factorizations 1-100
For each of the numbers you have been given, place that number’s “prime tower” on the appropriate location on the grid. Once this landscape has been constructed, do the following: 1
THINK: Identify at least three patterns you notice.
2
PAIR: Discuss what you noticed with a partner.
3
SHARE: Share one pattern with the large group.
Constructing Factor Lists Using Primes
Your instructor will assign your group a number: 96
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Using the prime factorization grid, highlight the cells of all the factors of that number. What patterns do you notice?
Constructing Factor Lists Using Primes
Constructing Factor Lists Using Primes
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
What It’s All About: The FTA
The Fundamental Theorem of Arithmetic (FTA) states that every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes (Hardy and Wright 1979).
What It’s All About: The FTA
FTA Consequence 1: Subset/Set comparisons of prime factorizations align with factor/multiple relationships
What It’s All About: The FTA
FTA Consequence 2: The greatest common factor of two numbers can be determined by finding the largest collection of prime factors that the two numbers share.
What It’s All About: The FTA
FTA Consequence 3: The least common multiple of two numbers can be determined by finding the smallest collection of prime factors that include both numbers.
Conclusion
Students can develop a richer understanding of the multiplicative structure of number through exposure to the “fundamental” role that prime numbers play.
Thank You! Matt Roscoe [email protected]
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Prime Quantity Quantity Required To Produce 107 2 97 53 3 48 26 5 24 18 7 16 10 11 9 8 13 7 6 17 5 6 19 5 5 23 4 4 29 3 4 31 3 3 37 2 3 41 2 3 43 2 3 47 2 2 53 1 2 59 1 2 61 1 2 67 1 2 71 1 2 73 1 2 79 1 2 83 1 2 89 1 2 97 1 239 279
Group N1 N2 N3 N4 N5 N6 N7 N8 N9
Prime Prime Multiple Multiple Relatively Prime Extra Extra Extra Extra
1 97 61 30 60 77 42 56 81 93
2 31 67 32 64 45 44 57 82 94
3 37 71 33 66 40 46 58 84 95
4 41 73 34 68 63 48 62 86 96
5 43 79 35 70 75 49 65 87 98
6 47 83 36 72 55 50 69 88 99
7 53 89 38 76 51 52 74 90 100
8 59 91 39 78 85 54 80 92 28
Ziv Feldman Boston University [email protected]
NCTM Research Conference San Francisco, CA April 15, 2016
1
Rationale
2
Setting the Stage
3
Explore the Landscape of Prime Factorizations 1-100
4
Constructing Factor Lists Using Primes
5
What It’s All About: The FTA
6
Conclusion
Rationale
“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” — G. Hardy (from A Mathematician’s Apology, London 1941).
Rationale
“...in mathematics, the primary subject-matter is not the mathematical objects but rather the structures in which they are arranged” (Resnik, 1999, p.201).
Rationale
Common Core State Standards for Mathematics (2010) Operations and Algebraic Thinking (4.OA) 4. Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite. (p. 29)
Rationale
Common Core State Standards for Mathematics (2010) Operations and Algebraic Thinking (4.OA) 4. Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite. (p. 29)
Rationale
A richer understanding of factors and multiples can be developed by drawing attention to the important role that prime numbers play when two numbers are multiplicatively compared.
Setting the Stage
Any whole number greater than zero with exactly two distinct, positive divisors is a prime number, or simply prime. Examples: 2, 3, 5, 7, 11
Setting the Stage
Any whole number greater than zero that has a positive factor other than 1 and itself is a composite number, or simply composite. Examples: 4, 6, 8, 10, 12
Setting the Stage The Sieve of Eratosthenes 1
Create a list of consecutive whole numbers starting from 2 to any n
2
The first prime is 2, let p “ 2
3
Circle p. Starting from p count up in multiples of p and eliminate each of these numbers (i.e. 2p, 3p, 4p, ....) until you reach n
4
Next, find the first number greater than p in the list that is not marked; let p equal this number and repeat the previous step
5
Continue this process until the list is exhausted, the circled numbers are the prime numbers up to n
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79
80
61
62
63
64
65
66
67
68
69
70
51
52
53
54
55
56
57
58
59
60
41
42
43
44
45
46
47
48
49
50
31
32
33
34
35
36
37
38
39
40
21
22
23
24
25
26
27
28
29
30
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
91
92
93
94
95
96
97
98
99 100
81
82
83
84
85
86
87
88
89
90
71
72
73
74
75
76
77
78
79
80
61
62
63
64
65
66
67
68
69
70
51
52
53
54
55
56
57
58
59
60
41
42
43
44
45
46
47
48
49
50
31
32
33
34
35
36
37
38
39
40
21
22
23
24
25
26
27
28
29
30
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
91
92
93
94
95
96
97
98
99 100
81
82
83
84
85
86
87
88
89
90
71
72
73
74
75
76
77
78
79
80
61
62
63
64
65
66
67
68
69
70
51
52
53
54
55
56
57
58
59
60
41
42
43
44
45
46
47
48
49
50
31
32
33
34
35
36
37
38
39
40
21
22
23
24
25
26
27
28
29
30
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
91
92
93
94
95
96
97
98
99 100
81
82
83
84
85
86
87
88
89
90
71
72
73
74
75
76
77
78
79
80
61
62
63
64
65
66
67
68
69
70
51
52
53
54
55
56
57
58
59
60
41
42
43
44
45
46
47
48
49
50
31
32
33
34
35
36
37
38
39
40
21
22
23
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25
26
27
28
29
30
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
91
92
93
94
95
96
97
98
99 100
81
82
83
84
85
86
87
88
89
90
71
72
73
74
75
76
77
78
79
80
61
62
63
64
65
66
67
68
69
70
51
52
53
54
55
56
57
58
59
60
41
42
43
44
45
46
47
48
49
50
31
32
33
34
35
36
37
38
39
40
21
22
23
24
25
26
27
28
29
30
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
91
92
93
94
95
96
97
98
99 100
81
82
83
84
85
86
87
88
89
90
71
72
73
74
75
76
77
78
79
80
61
62
63
64
65
66
67
68
69
70
51
52
53
54
55
56
57
58
59
60
41
42
43
44
45
46
47
48
49
50
31
32
33
34
35
36
37
38
39
40
21
22
23
24
25
26
27
28
29
30
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
91
92
93
94
95
96
97
98
99 100
81
82
83
84
85
86
87
88
89
90
71
72
73
74
75
76
77
78
79
80
61
62
63
64
65
66
67
68
69
70
51
52
53
54
55
56
57
58
59
60
41
42
43
44
45
46
47
48
49
50
31
32
33
34
35
36
37
38
39
40
21
22
23
24
25
26
27
28
29
30
11
12
13
14
15
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17
18
19
20
1
2
3
4
5
6
7
8
9
10
Setting the Stage
Factor Tree Method
Setting the Stage
Factor Lattice Method 72
Setting the Stage
Factor Lattice Method
2
72 36
Setting the Stage
Factor Lattice Method
2 2
72 36 18
Setting the Stage
Factor Lattice Method
2 2 2
72 36 18 9
Setting the Stage
Factor Lattice Method
2 2 2 3
72 36 18 9 3
Setting the Stage
Factor Lattice Method
2 2 2 3
72 36 18 9 3
Explore the Landscape of Prime Factorizations 1-100
You each are given a collection of numbers. Do the following: 1
2
Construct each number as a product of primes and find a collection of blocks to represent the number and construct a “prime tower”. Once you have constructed all your towers, discuss the following questions: Which pairs of numbers share factors? Which pairs do not? How do you know? 2 How can you find the largest factor that a pair of numbers share? How do you know? 3 Are there any pairs where one number is a multiple of the other? How do you know? 1
Explore the Landscape of Prime Factorizations 1-100
For each of the numbers you have been given, place that number’s “prime tower” on the appropriate location on the grid. Once this landscape has been constructed, do the following: 1
THINK: Identify at least three patterns you notice.
2
PAIR: Discuss what you noticed with a partner.
3
SHARE: Share one pattern with the large group.
Constructing Factor Lists Using Primes
Your instructor will assign your group a number: 96
90
84
80
78
72
Using the prime factorization grid, highlight the cells of all the factors of that number. What patterns do you notice?
Constructing Factor Lists Using Primes
Constructing Factor Lists Using Primes
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
Constructing Factor Lists Using Primes “Seeing” all the factors of 70.
What It’s All About: The FTA
The Fundamental Theorem of Arithmetic (FTA) states that every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes (Hardy and Wright 1979).
What It’s All About: The FTA
FTA Consequence 1: Subset/Set comparisons of prime factorizations align with factor/multiple relationships
What It’s All About: The FTA
FTA Consequence 2: The greatest common factor of two numbers can be determined by finding the largest collection of prime factors that the two numbers share.
What It’s All About: The FTA
FTA Consequence 3: The least common multiple of two numbers can be determined by finding the smallest collection of prime factors that include both numbers.
Conclusion
Students can develop a richer understanding of the multiplicative structure of number through exposure to the “fundamental” role that prime numbers play.
Thank You! Matt Roscoe [email protected]
91
92
93
94
95
96
97
98
99
100
81
82
83
84
85
86
87
88
89
90
71
72
73
74
75
76
77
78
79
80
61
62
63
64
65
66
67
68
69
70
51
52
53
54
55
56
57
58
59
60
41
42
43
44
45
46
47
48
49
50
31
32
33
34
35
36
37
38
39
40
21
22
23
24
25
26
27
28
29
30
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
91
92
22 ·23
7·13 81
34 71
51
41
41 31
21
3·7
22
11
12
1
1
23
2
2
3
3
24
17
6
2·3
39
23
30
2·3·5 19
20
22 ·5
19 9
32
40
23 ·5
29
8
50
2·52
29
2·32 7
7
49
18
60
22 ·3·5
3·13
22 ·7
17
59
28
70
2·5·7
72
2·19
33
3·23
38
27
16
5
5
37
69
59
24 ·3
80
24 ·5
79
48
37
26
2·13
3·5 4
22
22 ·32
15
2·29
47
79
58
47
36
25
14
2·7
46
35
52
57
90
2·32 ·5
89
68
22 ·17
3·19
2·23
5·7
23 ·3
13
32 ·5
24
13
23 ·7
45
34
2·17
23
22 ·3
11
33
3·11
2·11
44
7·11 2·3·13
56
89
78
67
100
32 ·11 22 ·52
23 ·11
67
99
88
77
66
55
5·11
22 ·11
43
65
54
43
76
5·13 2·3·11
2·33
53
32
25
26
87
3·29
22 ·19
98
2·72
97
2·43
3·52
97
86
75
64
53
42
2·3·7
31
63
32 ·7
22 ·13
85
74
2·37
96
25 ·3
5·19
22 ·3·7 5·17
73
52
95
84
73
62
2·31
3·17
83
72
61
94
2·47
83
23 ·32
61
3·31
82
2·41
71
93
10
2·5
1
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3
4
5
6
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8
9
10
11
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31
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37
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40
41
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43
44
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46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
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79
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81
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83
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86
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91
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99
100
91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50 31 32 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1
2
3
4
5
6
7
8
9
10
91
92
93
94
95
96
97
98
99
100
81
82
83
84
85
86
87
88
89
90
71
72
73
74
75
76
77
78
79
80
61
62
63
64
65
66
67
68
69
70
51
52
53
54
55
56
57
58
59
60
41
42
43
44
45
46
47
48
49
50
31
32
33
34
35
36
37
38
39
40
21
22
23
24
25
26
27
28
29
30
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
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
Prime Quantity Quantity Required To Produce 107 2 97 53 3 48 26 5 24 18 7 16 10 11 9 8 13 7 6 17 5 6 19 5 5 23 4 4 29 3 4 31 3 3 37 2 3 41 2 3 43 2 3 47 2 2 53 1 2 59 1 2 61 1 2 67 1 2 71 1 2 73 1 2 79 1 2 83 1 2 89 1 2 97 1 239 279
Group N1 N2 N3 N4 N5 N6 N7 N8 N9
Prime Prime Multiple Multiple Relatively Prime Extra Extra Extra Extra
1 97 61 30 60 77 42 56 81 93
2 31 67 32 64 45 44 57 82 94
3 37 71 33 66 40 46 58 84 95
4 41 73 34 68 63 48 62 86 96
5 43 79 35 70 75 49 65 87 98
6 47 83 36 72 55 50 69 88 99
7 53 89 38 76 51 52 74 90 100
8 59 91 39 78 85 54 80 92 28
