# 5.4: What's My Rule

Difficulty Level: At Grade Created by: CK-12

Look at the table below. \begin{align*}p\end{align*} and \begin{align*}t\end{align*} stand for numbers. Can you write a rule to represent the relationship between \begin{align*}p\end{align*} and \begin{align*}t\end{align*}? 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.

\begin{align*}& \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 \end{align*}

\begin{align*}u\end{align*} \begin{align*}v\end{align*}
7 9
4 6
1 3
0 2
1. 2 4
2. 9 11
3. 3 5
4. 8 10
5. Rule: \begin{align*}v=u+2\end{align*}

\begin{align*}& \mathbf{Check:} && 2+2=4\\ &&& 9+2=11\\ &&&3+2=5\\ &&&8+2=10\end{align*}

#### Example B

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

Solution:

We can use problem solving steps to help.

\begin{align*}& \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. \end{align*}

\begin{align*}w\end{align*} \begin{align*}y\end{align*}
10 6
15 11
13 9
5 1
1. 8 4
2. 14 10
3. 7 3
4. 16 12
5. Rule: \begin{align*}y=w-4\end{align*}

\begin{align*}& \mathbf{Check:} && 8-4=4\\ &&& 14-4=10\\ &&&7-4=3\\ &&&16-4=12\end{align*}

#### Example C

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

Solution:

We can use problem solving steps to help.

\begin{align*}& \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 \end{align*}

\begin{align*}z\end{align*} \begin{align*}a\end{align*}
7 37
10 40
18 48
45 75
1. 31 61
2. 14 44
3. 8 38
4. 29 59
5. Rule: \begin{align*}a=z+30\end{align*}

\begin{align*}& \mathbf{Check:} && 31+30=61\\ &&& 14+30=44\\ &&&8+30=38\\ &&&29+30=59\end{align*}

#### Concept Problem Revisited

We can use problem solving steps to help.

\begin{align*}& \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}\end{align*}

### 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:

\begin{align*}b\end{align*} \begin{align*}c\end{align*}
48 28
50 30
28 8
42 22
1. 67 47
2. 59 39
3. 75 55
4. 34 14
5. Rule: \begin{align*}c=b-20\end{align*}

2. Here is the completed table and rule:

\begin{align*}d\end{align*} \begin{align*}e\end{align*}
3 12
8 32
2 8
5 20
1. 1 4
2. 7 28
3. 6 24
4. 9 54
5. Rule: \begin{align*}e=4 \times d\end{align*}

3. Here is the completed table and rule:

\begin{align*}f\end{align*} \begin{align*}g\end{align*}
10 2
40 8
25 5
5 1
1. 15 3
2. 35 7
3. 20 4
4. 45 9
5. Rule: \begin{align*}g=f \div 5\end{align*}

4. Here is the completed table and rule:

\begin{align*}h\end{align*} \begin{align*}j\end{align*}
2 18
5 45
7 63
1 9
1. 4 36
2. 3 27
3. 6 54
4. 9 81
5. Rule: \begin{align*}j=9 \times h\end{align*}

### Practice

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

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Date Created:
Jan 18, 2013