Discovering Divisibility Rules:
Finding Patterns on the Hundreds Chart
Students will look for and express regularity in repeated reasoning
in order to create a divisibility rules for 2, 3, 4, 5, 6, 9, and 10.
Includes:
- Student Handouts
- Probing Questions to Guide Students
- Answer Key
Copyright © 2013 Gemma Dimery
All rights reserved by author.
Permission to copy for single classroom use only.
Electronic distribution limited to single classroom use only.
Not for public display.
Discovering Divisibility Rules for 2, 3, & 6
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 1011 / 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 / 50
51 / 52 / 53 / 54 / 55 / 56 / 57 / 58 / 59 / 60
61 / 62 / 63 / 64 / 65 / 66 / 67 / 68 / 69 / 70
71 / 72 / 73 / 74 / 75 / 76 / 77 / 78 / 79 / 80
81 / 82 / 83 / 84 / 85 / 86 / 87 / 88 / 89 / 90
91 / 92 / 93 / 94 / 95 / 96 / 97 / 98 / 99 / 100
Color the multiples of 2 in yellow.
What patterns do you notice? ______
What rule can we write to know if a number is divisible by 2? ______
______
Outline the multiples of 3 in red.
What patterns do you notice? ______
What rule can we write to know if a number is divisible by 3? ______
______
Circle the multiples of 6.
What patterns do you notice? ______
What rule can we write to know if a number is divisible by 3? ______
______
Discovering Divisibility Rules for 510
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 1011 / 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 / 50
51 / 52 / 53 / 54 / 55 / 56 / 57 / 58 / 59 / 60
61 / 62 / 63 / 64 / 65 / 66 / 67 / 68 / 69 / 70
71 / 72 / 73 / 74 / 75 / 76 / 77 / 78 / 79 / 80
81 / 82 / 83 / 84 / 85 / 86 / 87 / 88 / 89 / 90
91 / 92 / 93 / 94 / 95 / 96 / 97 / 98 / 99 / 100
Color the multiples of 5 in blue.
What patterns do you notice? ______
What rule can we write to know if a number is divisible by 5? ______
______
Outline the multiples of10 in green.
What patterns do you notice? ______
What rule can we write to know if a number is divisible by 10? ______
______
Discovering Divisibility Rules for 49
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 1011 / 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 / 50
51 / 52 / 53 / 54 / 55 / 56 / 57 / 58 / 59 / 60
61 / 62 / 63 / 64 / 65 / 66 / 67 / 68 / 69 / 70
71 / 72 / 73 / 74 / 75 / 76 / 77 / 78 / 79 / 80
81 / 82 / 83 / 84 / 85 / 86 / 87 / 88 / 89 / 90
91 / 92 / 93 / 94 / 95 / 96 / 97 / 98 / 99 / 100
Outline the multiples of 9 in purple.
What patterns do you notice? ______
What rule can we write to know if a number is divisible by 9? ______
______
Color the multiples of4 in pink.
What patterns do you notice? ______
What rule can we write to know if a number is divisible by 4? ______
______
Discovering Divisibility Rules for 2, 3, & 6
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 1011 / 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 / 50
51 / 52 / 53 / 54 / 55 / 56 / 57 / 58 / 59 / 60
61 / 62 / 63 / 64 / 65 / 66 / 67 / 68 / 69 / 70
71 / 72 / 73 / 74 / 75 / 76 / 77 / 78 / 79 / 80
81 / 82 / 83 / 84 / 85 / 86 / 87 / 88 / 89 / 90
91 / 92 / 93 / 94 / 95 / 96 / 97 / 98 / 99 / 100
Creating a Divisibility Rule for 2
Question 1: What patterns do you notice?
The tens increase by 1.
The ones are the same in each column.
The numbers are even.
Question 2: Do our patterns work for just the yellow numbers? If, the pattern can be used for the other numbers that are not yellow this is not a good pattern to use to create a rule. Our rule has to work for just the yellow numbers
The tens increase by 1
Is this only true for the yellow numbers? No, 65,75,85… This pattern does not help us create the rule.
The ones are the same in each column
Is this only true for the yellow numbers? No-79,89,99… This does not help us create the rule.
The numbers are even
Is this only true for the yellow numbers?Yes, this will help us create a rule.
Question 3: What rule can we write to know if a number is divisible by 2?
Rule: A number is divisible by 2 if it is even
Creating a Divisibility Rule for 3
Question 1: What patterns do you notice?
The numbers are diagonal
“What do you mean? “
“How do to the numbers change in the diagonal?”(Tens increase, ones decrease) add 10 subtract 1
Each column jumps by 30
What would come after 99 in the diagonal pattern? Explain.(108, tens increase by one and ones decrease by 1, 90 become 100 and 9 becomes 8; or 78 + 30 = 108 if you follow the pattern in the column)
Question 2: Do our patterns work for just the red outlined numbers? If, the pattern can be used for the other numbers that are not outlined this is not a good pattern to use to create a rule. Our rule has to work for just the red numbers
Diagonals and Column jumps by 30 does not help because we have to have a starting point that is divisible by 3 for that to work.
Question 3: “What do you notice if we find the sum of the digits of the red numbers?”
The diagonals have the same sums; the sums are multiples of 3
Does this help us create our rule? (Yes)
How do you know? The other sum of the digits of the other numbers are not multiples of 3; Example: 95 9+5=14, 14 is not a multiple of 3)
Question 4: What rule can we write to know if a number is divisible by 3?
Rule:If the sum of the digits of the number is a multiple of 3, the number is divisible by 3.
Creating a Divisibility Rule for 6
Question 1: What patterns do you notice?
The numbers with x’s are even, the numbers with x’s are also red and yellow.
Question 2: How can this help us write our rule?
All the numbers that are divisible by 6 are colored yellow and red.No other numbers are circled.
What does it mean if the numbers are yellow? (divisible by 2)
What does it mean if the numbers are red? (divisible by 3)
Question 3: What rule could I write?
Rule: If a number is divisible by 2 and 3, the number is also divisible by 6.
Discovering Divisibility Rules for 510
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 1011 / 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 / 50
51 / 52 / 53 / 54 / 55 / 56 / 57 / 58 / 59 / 60
61 / 62 / 63 / 64 / 65 / 66 / 67 / 68 / 69 / 70
71 / 72 / 73 / 74 / 75 / 76 / 77 / 78 / 79 / 80
81 / 82 / 83 / 84 / 85 / 86 / 87 / 88 / 89 / 90
91 / 92 / 93 / 94 / 95 / 96 / 97 / 98 / 99 / 100
Creating a Divisibility Rule for 5
Question 1: What patterns do you notice?
The numbers end in 5 and zero, there is a 5 or a 0 in the ones place.
Question 2: How can this help us write our rule?
This pattern is true only for the blue numbers?
Question 3: What rule could I write?
Rule: A number is divisible by 5 if the number ends in 5 or zero
Creating a Divisibility Rule for 10
Question 1: What patterns do you notice?
The numbers end in zero, there is a 0 in the ones place.
Question 2: How can this help us write our rule?
This pattern is true only for the green outlined numbers?
Question 3: What rule could I write?
Rule: A number is divisible by 10 if the number ends in zero
Discovery Divisibility Rules for 49
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 1011 / 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 / 50
51 / 52 / 53 / 54 / 55 / 56 / 57 / 58 / 59 / 60
61 / 62 / 63 / 64 / 65 / 66 / 67 / 68 / 69 / 70
71 / 72 / 73 / 74 / 75 / 76 / 77 / 78 / 79 / 80
81 / 82 / 83 / 84 / 85 / 86 / 87 / 88 / 89 / 90
91 / 92 / 93 / 94 / 95 / 96 / 97 / 98 / 99 / 100
Creating a Divisibility Rule for 9
Question 1: What patterns do you notice?
The numbers form a diagonal pattern.
What do you mean? How do the numbers change? (add 10, subtract 1)
Question 2: How can this help us write our rule?
Diagonal does not help because we have to have a starting point that is divisible by 3 for that to work.
What do you notice about the sums of the digits of the purple numbers?
The sum of the digits is 9 and 18.
Is this true only for the purple numbers?
Yes (test a few of the other numbers to prove it)
What number would come after 99 in the diagonal pattern? 108. What is the sum of the digits? 9
Question 3: What rule could I write?
Rule: A number is divisible by 9 if the sum of the digits is a multiple of 9
Creating a Divisibility Rule for 4
Question 1: What patterns do you notice?
The numbers are even
The number jump by 20 in each column
Question 2: How can this help us write our rule?
The following numbers are divisible by "4":
4
8
12 4
168
20 12
16
Now, add "20" to each of those numbers: 20
24 Tens 1,1,2,2,2,3,3,4,4,4,5,5,
24 28
28 32 Ones 4,8,2,6,0,4,8,2,6,0,4,8,2,6
32 36
36 40
40 44
48
Again, add "20" to each of them. 52
56
44 60
48
52
56
60
Question 3: What rule could I write?
Rules:
Numbers with an odd number in the ten’s place will be divisible by 4 if the number in their one's place is either 2 or 6
Numbers with an even digit in the ten's place will be divisible by 4 if the number in their one's place is either 0, 4, or 8.
A general rule for this can be stated that if the last two digits of a number are divisible by 4, then the number is also divisible by 4
Copyright © 2013 Gemma Dimery