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# 5.4: What's My Rule

Created by: CK-12

Look at the table below. $p$ and $t$ stand for numbers. Can you write a rule to represent the relationship between $p$ and $t$ ? Can you complete the table? In this concept we will learn how to write rules to represent what we see in tables.

### Guidance

In order to write rules for function tables like the one above, we can use the problem solving steps to help.

• First, describe what you see in the table.
• Second, identify what your job is. In these problems, your job will be to write the rule for the table.
• Third, make a plan . In these problems, your plan should be to look for a pattern to help you figure out the rule.
• Fourth, solve . In these problems, you will state the rule in an equation or in words and then complete the table.
• Fifth, check . In these problems you will check by making sure that the numbers in the table work with the rule.

#### Example A

Write the rule represented by the table. Then, complete the table.

Solution:

We can use problem solving steps to help.

$& \mathbf{Describe} && \text{I see a table with two columns}.\\ &&& u \ \text{and} \ v \ \text{stand for numbers}.\\ & \mathbf{My \ Job} && \text{Figure out the rule}.\\&&& \text{What do I have to do to} \ u \ \text{to get} \ v?\\ &&& \text{Then complete rows} \ 1, 2, 3 \ \text{and} \ 4.\\ & \mathbf{Plan} && \text{I can add, subtract, multiply or divide. I have to use the same operation on}\\&&& \text{all} \ u \ \text{numbers to get the} \ v \ \text{numbers}.\\& \mathbf{Solve} && \text{All} \ v \ \text{numbers are} \ 2 \ \text{more than } \ u \ \text{numbers}.\\&&& \text{The rule is add 2 to } \ u \ \text{to get} \ v. \ \text{I can write} \ v=u+2$

$u$ $v$
7 9
4 6
1 3
0 2
1. 2 4
2. 9 11
3. 3 5
4. 8 10
5. Rule: $v=u+2$

$& \mathbf{Check:} && 2+2=4\\&&& 9+2=11\\&&&3+2=5\\&&&8+2=10$

#### Example B

Write the rule represented by the table. Then, complete the table.

Solution:

We can use problem solving steps to help.

$& \mathbf{Describe} && \text{I see a table with two columns}.\\ &&& w \ \text{and} \ y \ \text{stand for numbers}.\\ & \mathbf{My \ Job} && \text{Figure out the rule}.\\&&& \text{What do I have to do to} \ w \ \text{to get} \ y?\\ &&& \text{Then complete rows} \ 1, 2, 3 \ \text{and} \ 4.\\ & \mathbf{Plan} && \text{I can add, subtract, multiply or divide. I have to use the same operation on}\\&&& \text{all} \ w \ \text{numbers to get the} \ y \ \text{numbers}.\\& \mathbf{Solve} && \text{All} \ y \ \text{numbers are} \ 4 \ \text{less than } \ w \ \text{numbers}.\\&&& \text{The rule is subtract 4 from } \ w \ \text{to get} \ y. \ \text{I can write} \ y=w-4.$

$w$ $y$
10 6
15 11
13 9
5 1
1. 8 4
2. 14 10
3. 7 3
4. 16 12
5. Rule: $y=w-4$

$& \mathbf{Check:} && 8-4=4\\&&& 14-4=10\\&&&7-4=3\\&&&16-4=12$

#### Example C

Write the rule represented by the table. Then, complete the table.

Solution:

We can use problem solving steps to help.

$& \mathbf{Describe} && \text{I see a table with two columns}.\\ &&& z \ \text{and} \ a \ \text{stand for numbers}.\\ & \mathbf{My \ Job} && \text{Figure out the rule}.\\&&& \text{What do I have to do to} \ z \ \text{to get} \ a?\\ &&& \text{Then complete rows} \ 1, 2, 3 \ \text{and} \ 4.\\ & \mathbf{Plan} && \text{I can add, subtract, multiply or divide. I have to use the same operation on}\\&&& \text{all} \ p \ \text{numbers to get the} \ t \ \text{numbers}.\\& \mathbf{Solve} && \text{All} \ a \ \text{numbers are} \ 30 \ \text{more than } \ z \ \text{numbers}.\\&&& \text{The rule is add 30 to } \ z \ \text{to get} \ a. \ \text{I can write} \ a=z+30$

$z$ $a$
7 37
10 40
18 48
45 75
1. 31 61
2. 14 44
3. 8 38
4. 29 59
5. Rule: $a=z+30$

$& \mathbf{Check:} && 31+30=61\\&&& 14+30=44\\&&&8+30=38\\&&&29+30=59$

#### Concept Problem Revisited

We can use problem solving steps to help.

$& \mathbf{Describe} && \text{I see a table with two columns}.\\ &&& p \ \text{and} \ t \ \text{stand for numbers}.\\ & \mathbf{My \ Job} && \text{Figure out the rule}.\\&&& \text{What do I have to do to} \ p \ \text{to get} \ t?\\ &&& \text{Then complete rows} \ 1, 2, 3 \ \text{and} \ 4.\\ & \mathbf{Plan} && \text{I can add, subtract, multiply or divide. I have to use the same operation on}\\&&& \text{all} \ p \ \text{numbers to get the} \ t \ \text{numbers}.\\& \mathbf{Solve} && \text{All} \ t \ \text{numbers are} \ 3 \ \text{times} \ p \ \text{numbers}.\\&&& \text{The rule is multiply} \ p \ \text{by} \ 3 \ \text{to get} \ t. \ \text{I can write} \ t = 3 \times p\\& \mathbf{Check}$

### Vocabulary

A function is an example of a rule that states a relationship between two quantities. A variable is a letter that represents a number that can change value. In this concept, we wrote rules that stated how the two variables in the table were related.

### Guided Practice

Write the rule represented by each table. Then, complete each table.

1.

2.

3.

4.

1. Here is the completed table and rule:

$b$ $c$
48 28
50 30
28 8
42 22
1. 67 47
2. 59 39
3. 75 55
4. 34 14
5. Rule: $c=b-20$

2. Here is the completed table and rule:

$d$ $e$
3 12
8 32
2 8
5 20
1. 1 4
2. 7 28
3. 6 24
4. 9 54
5. Rule: $e=4 \times d$

3. Here is the completed table and rule:

$f$ $g$
10 2
40 8
25 5
5 1
1. 15 3
2. 35 7
3. 20 4
4. 45 9
5. Rule: $g=f \div 5$

4. Here is the completed table and rule:

$h$ $j$
2 18
5 45
7 63
1 9
1. 4 36
2. 3 27
3. 6 54
4. 9 81
5. Rule: $j=9 \times h$

### Practice

Write the rule represented by each table. Then, complete each table.

Jan 18, 2013