# Patterns in Division Contributed by: This includes Divisibility which refers to whether or not a number is divisible by another number. If a number divides evenly into another number (no remainder), then it is divisible by that number.
1. Grade 7 Mathematics Unit 1: Patterns and Relations
Section 1.1 – Patterns in Division
Divisibility refers to whether or not a number is divisible by another number.
If a number divides evenly into another number (no remainder), then it is
divisible by that number.
For example, . 36 is divisible by 4 since 9 divides evenly into 36 (there is
no remainder).
Divisibility by 10
Consider the following numbers. Circle the numbers that
are divisible by 10.
44 50 62
75 90 38 40
10 88 120
How do we know if a number is divisible by 10?
Rule: __________________________________________________________________________________________
Divisibility by 2
Consider the following numbers. Circle the numbers that are divisible by 2.
34 99 59
52 78 67 32
52 46 31
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2. Grade 7 Mathematics Unit 1: Patterns and Relations
How do we know if a number is divisible by 2?
Rule: __________________________________________________________________________________________
Divisibility by 5
Consider the following numbers. Circle the numbers that
are divisible by 5.
80 49 61
25 40 57 55
78 10 15
How do we know if a number is divisible by 5?
Rule: __________________________________________________________________________________________
Example 1:
Circle the numbers that are divisible by both 2 and by 5.
54 20 33
75 40 48 65
22 10 15
Venn diagrams are diagrams that use circles to represent groups and to show the
relationship between the groups.
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3. Grade 7 Mathematics Unit 1: Patterns and Relations
We can use a Venn Diagram to show the numbers divisible by 2 and 5.
Divisible Divisible
by 2 by 5
Divisible
by 2 AND 5
Note: A number that is not divisible by either number is placed on the outside of the
Divisibility by 4 and 8
Trying to figure out which numbers are divisible by 4 and 8 can be a
little more difficult. However, we can develop a rule to help us
quickly figure it out without having to complete long division.
On the hundreds chart below, place a circle around all the numbers
divisible by 4.
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4. Grade 7 Mathematics Unit 1: Patterns and Relations
On the chart below, continue the pattern.
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5. Grade 7 Mathematics Unit 1: Patterns and Relations
What do you notice? Write a rule for numbers divisible by 4.
Rule: __________________________________________________________________________________________
Which of the following numbers is divisible by 4? Justify your answer.
24 321 436 2048
Example 2:
Using the digits 0-9, replace the  in each number with all the possibilities that will
make each number divisible by 4.
a) 13
b) 148
c) 234
d) 1552
Divisibility by 8 is similar to the rule for divisibility by 4 but, instead of the
last two digits, we look at the last three.
Rule: A number is divisible by 8 if the last three digits are divisible by 8.
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6. Grade 7 Mathematics Unit 1: Patterns and Relations
Example 1:
Explain which of the following are divisible by 8. Show how you know.
a) 5872
b) 12 168
c) 3 024
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7. Grade 7 Mathematics Unit 1: Patterns and Relations
Section 1.2 – More Patterns in Division
Divisibility by 3
Complete the chart below. The first one is done for you.
Number Divisible by 3? Sum of Digits Sum of Digits Divisible by 3?
30 yes 9+0=9 yes
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
What do you notice??
Rule: __________________________________________________________________________________________
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8. Grade 7 Mathematics Unit 1: Patterns and Relations
Divisibility by 9
Complete the chart below. The first one is done for you
Number Divisible by 9? Sum of Digits Sum of Digits Divisible by 9?
18 yes 9+0=9 Yes
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
What do you notice?
Rule: _________________________________________________________________________________________
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9. Grade 7 Mathematics Unit 1: Patterns and Relations
Divisibility by 6
Sort the following numbers and place them in the Venn diagram.
12, 21, 36, 42, 56, 61, 74, 88, 93, 135, 246, 453, 728
Divisible by 2 Divisible by 3 Divisible by Neither
Let’s place the numbers in a Venn diagram!
What do you notice about the numbers in the overlapping region of the diagram?
Write a rule for divisibility by 6:
Rule: __________________________________________________________________________________________
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10. Grade 7 Mathematics Unit 1: Patterns and Relations
We can also sort numbers using a Carroll diagram.
Divisible by 2 Not Divisible by 2
Divisible by 3
Not Divisible by 3
Let’s sort these numbers: 1, 11, 15, 20, 24, 35, 47, 98, 100
Divisible by 5 Not Divisible by 5
Divisible by 2
Not Divisible by 2
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11. Grade 7 Mathematics Unit 1: Patterns and Relations
Section 1.3 – Algebraic Expressions
Joe makes \$3 on every chocolate bar he sells. How much money he earns,
depends on how many bars he sells each week.
We can express this situation as 3b.
 This means 3×b (3 times b, since he gets \$3 per chocolate bar)
 “b” represents the number of chocolate bars
 “b” is called a variable
3b is called an expression.
In the expression 3b, 3 is called the numerical coefficient.
Numerical coefficient: _____________________________________________________________________
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12. Grade 7 Mathematics Unit 1: Patterns and Relations
In the expression 3b + 5, 5 is called the constant.
Constant Term: _____________________________________________________________________________
Example 1:
In each expression, identify the variable, numerical coefficient and constant term.
a) 3r+7
Variable: ____________
Numerical Coefficient: ____________
Constant Term: ____________
b) 4h – 1.3
Variable: ____________
Numerical Coefficient: ____________
Constant Term: ____________
c) 19 – 6w
Variable: ____________
Numerical Coefficient: ____________
Constant Term: ____________
d) ½ d+3
Variable: ____________
Numerical Coefficient: ____________
Constant Term: ____________
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13. Grade 7 Mathematics Unit 1: Patterns and Relations
e) 5.4k
Variable: ____________
Numerical Coefficient: ____________
Constant Term: ____________
f) c – 8
Variable: ____________
Numerical Coefficient: ____________
Constant Term: ____________
Example 2:
Write expressions for the following:
a) Five more than a number _________________________________________________
b) Three less than a number __________________________________________________
c) Six times a number __________________________________________________________
d) Three more than two times a number __________________________________________________
e) A number divided by twenty _____________________________________________________________
f) One hundred divided by a number ______________________________________________________
g) Seven subtracted from four times a number ___________________________________________
h) Twelve time a number is added to fifteen ______________________________________________
i) Nine more than triple a number ________________________________________________________
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14. Grade 7 Mathematics Unit 1: Patterns and Relations
Just as we can write expressions for sentences, we can write sentences (words) for
Example 3:
a) 13p ____________________________________________________________________________________
b) m + 12 _________________________________________________________________________________
c) p/2 _____________________________________________________________________________________
d) 3k + 6 __________________________________________________________________________________
e) 16 – n/2 _______________________________________________________________________________
We can evaluate an expression for a given value, by “plugging” a value in where you
see the variable.
Example 4:
Evaluate each expression assuming that n = 4.
a) 4n = 4 (4) = 16
b) 12/n =
c) 14 – n =
d) n + 8 =
e) 2n + 7 =
f) 28 – 24/n =
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15. Grade 7 Mathematics Unit 1: Patterns and Relations
Worksheet
Part A: What words match with the mathematical operations?
Math Symbol Words
+
-
÷
x
Part B: Write English phrases for the following mathematical expressions.
Expression Sentence
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16. Grade 7 Mathematics Unit 1: Patterns and Relations
Part C: Translate each English expression or equation into mathematical form.
English Expression
1. Double a number
2. A number increased by six.
3. A number decreased by four.
4. The sum of a number and ten.
5. Seven times a number.
6. Seven less than a number.
7. Half of a number increased by nine.
8. A number increased by seven is fourteen.
9. Three times a number plus six is twenty-four.
10. One-quarter of a number equals eighteen.
11. A number divided by five and then decreased by
eight.
12. Seven decreased by a number
Question: Which of the above are equations? How do you know?
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17. Grade 7 Mathematics Unit 1: Patterns and Relations
Part D: For each algebraic expression, identify the numerical coefficient, the
variable, and the constant term.
Algebraic Numerical
Variable Constant Term
Expression Coefficient
Part E: Evaluate each expression by replacing the variable with the given number.
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18. Grade 7 Mathematics Unit 1: Patterns and Relations
Section 1.4 –Relationships in Patterns
Consider the pattern:
1 2 3 4
We can show this in a table:
Diagram #
1 2 3 4 5
(d)
# of circles
(c)
What do you notice?
What relationship do you see between the diagram number and the number of dots?
In Words:
3d is a _________________________________________ because the variable, _____________, is
related to the number of circles and vice versa.
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19. Grade 7 Mathematics Unit 1: Patterns and Relations
How is each term related to the term number? Write a relation for each.
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20. Grade 7 Mathematics Unit 1: Patterns and Relations
Section 1.5 –Patterns and Relationships in Tables
We can represent a relation using an input/output table. We enter numbers in the
input column, do what the relation tells us, and write the result under the output.
These tables come in handy when we want to graph our relations.
Examples: Complete each table and explain how the output is related to the input.
The output is two more than three times the input.
The output is twelve minus the input.
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21. Grade 7 Mathematics Unit 1: Patterns and Relations
The output is three more than 5 times the input
We can also write the relation using algebra when we are given the table.
Examples: Write a relation for each table.
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22. Grade 7 Mathematics Unit 1: Patterns and Relations
Section 1.6 – Graphing Relations
We can use graphs to show the relationship between two quantities.
Consider the example below:
1. Triangles are used to create the pattern below:
Diagram 1 Diagram 2 Diagram 3
Complete the table and graph the relation:
(n) Triangles (t)
1
2
3
4
5
6
** Remember, the input goes on the bottom
(horizontal axis) and the output goes on the side
(vertical axis).
Write a relation to show how the
number of squares is related to the diagram number, n.
When the points on a graph fall in a straight line, the relation is called a
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23. Grade 7 Mathematics Unit 1: Patterns and Relations
Try the following examples:
2. Sheila was having a party and could arrange the table and chairs as follows:
Complete the table and graph the relation.
Number of Number of
tables (n) people
1
2
3
4
5
Write a relation to show how the number
of squares is related to the diagram number, n.
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24. Grade 7 Mathematics Unit 1: Patterns and Relations
3. Square tiles are used to create the pattern below.
Complete the table and graph the relation.
Diagram Number of
number(n) squares
1
2
3
4
5
Write a relation to show how the number
of squares is related to the diagram number, n.
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25. Grade 7 Mathematics Unit 1: Patterns and Relations
Section 1.7 –Reading and Writing Equations
Let’s be math detectives!!
Example 1:
I am thinking of a number. If you multiply it by 3 and add 4,
you will get 13. What is the number?
(Clue: Write the algebraic equation first!)
What is an algebraic equation?
It is a _________________________________________________________________ describing the
relationship , using an _______________________________________, between two expressions.
, then = __________
, then = __________
Example 2:
I am thinking of a number. If you multiply it by 5 and subtract 4, the answer is 21.
What is the number?
Example 3:
Katelyn bought 3 CD's. Each CD cost the same amount. The total cost is \$36.00.
A) Write the algebraic equation
B) What is the cost of one CD?
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26. Grade 7 Mathematics Unit 1: Patterns and Relations
Example 4:
Write an equation for each sentence:
A) Three more than a number is 15.
B) A number subtracted from 5, is 1.
C) Eight added to three times a number is 26.
Example 5:
Write a sentence for each equation:
A)
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27. Grade 7 Mathematics Unit 1: Patterns and Relations
B)
C)
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28. Grade 7 Mathematics Unit 1: Patterns and Relations
Section 1.8 – Solving Equations Using Algebra Tiles
We can use algebra tiles to represent an expression or an equation.
= +1, called a unit tile.
= x, called a variable tile.
a) can be represented as
What expression is represented by the following?
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29. Grade 7 Mathematics Unit 1: Patterns and Relations
We can solve algebraic expressions using tiles:
For example:
We want to get the variable tiles (long tiles) on one side by themselves.
To do this, we take away 3 tiles on the left side....but to keep the
equation balanced, we must take away three tiles on the right side
Example: Use tiles to solve each:
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30. Grade 7 Mathematics Unit 1: Patterns and Relations
= = so, =
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31. Grade 7 Mathematics Unit 1: Patterns and Relations
If we have , we want .
So, we need __________________________________________________to complete a whole.
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