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# Input-Output Tables

## Use patterns to find missing values in tables.

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Input-Output Tables

Sonja’s 4-H club has spent the last two months collecting food donations for their community service project. The club plans to distribute food bags to needy families during the thanks giving holiday. Sonja is in charge of preparing the food bags. She has enlisted the help of several of her club members. The first day, Sonja and two other club members prepared 9 bags. The second day, Sonja prepared 3 bags by herself.  On the third day, Angela helped Sonja and the two of them prepared 6 bags. On the fourth day, five members showed up to help Sonja and the six of them prepared 18 bags. Sonja organized this data into and input-out table.

 Input Output 1 3 2 6 3 9 6 18

Do you see a pattern? How can Sonja use the information in the table to find a pattern rule for preparing the bags?

In this concept, you will learn to write an expression for an input-output table.

### Writing Expressions for Input-Output Tables

A pattern is something that repeats in a specific way. Patterns are everywhere in life. They exist in nature and in machinery and even in temperatures. Detecting patterns is one of the things that mathematicians and scientists do every day. They look for patterns in the way that things are created or counted and then they can draw conclusions based on those patterns.

A pattern functions according to a rule. The rule tells us how the pattern repeats.

This group of numbers is a pattern.

2, 4, 6, 8, 10. . . . .

Once you have a pattern, you can establish a rule about the pattern. This pattern counts by two’s. You could also say that two is added to each previous term to get the next term in the pattern.

In this example, you could use a variable to represent the terms in the list. Let’s use \begin{align*}x\end{align*}.

\begin{align*}x = \text{term in the pattern}\end{align*}

The word “term” refers to the numbers 2, 4, 6 and so on.

Next, you can add more to the variable. Since you add two to each term to get the next term, then you can say that \begin{align*}x\end{align*} plus two is the rule.

\begin{align*}\text{Rule}: x + 2\end{align*}

Now let’s check the rule to be sure that it works for each term in the list. 2, 4, 6, 8, 10 . . .

If you take 2 and substitute it for \begin{align*}x\end{align*} then \begin{align*}2 + 2 = 4\end{align*}, so the rule works. If you take 4 and substitute it for \begin{align*}x\end{align*} then \begin{align*}4 + 2 = 6\end{align*}, so the rule works. If you take 6 and substitute it for \begin{align*}x\end{align*} then \begin{align*}6 + 2 = 8\end{align*}, so the rule works.

An easier way to figure this out would be to use a table called an input/output table.

 Input Output 2 4 4 6 6 8 8 10

Let’s see if the rule \begin{align*}x + 2\end{align*} works for this table.

A term has been put into the table; that is the input. Then a term comes out; that is the output. The rule tells us what happened to the input to equal the output.

Two can be added to each term in the input column to equal the output column so the rule \begin{align*}x + 2\end{align*} works.

You can write rules by examining the patterns in input/output tables.

 Input Output 0 0 1 3 2 6 3 9

Figuring out a rule is a little like deciphering a puzzle. You have to think of what happened to one term to equal another term.

In this table, the term in the input column was multiplied by 3 to get the number in the output column.

You can write the rule as an expression.

If the input column is \begin{align*}x\end{align*}, then \begin{align*}3x\end{align*} is the rule for this table.

\begin{align*}\text{Rule} = 3x\end{align*}

Sometimes rules are a bit more complicated. Sometimes, there can be two operations in a rule.

 Input Output 3 7 4 9 5 11 7 15

Look for a pattern in this table.

You should see that the input was multiplied by two and then one was added to get the output. You can write the rule as an expression, using a variable as the input.

\begin{align*}\text{Rule} = 2x + 1\end{align*}

The input-output relationship of terms is a function, which is when one variable or terms depends on another according to a rule.

### Examples

#### Example 1

Earlier, you were given a problem about Sonja and her 4-H club’s community service project.

How can Sonja use the information presented in the table below to come up with a pattern rule that explains how the number of people preparing bags affects the output or number of bags prepared?

 Input Output 1 3 2 6 3 9 6 18

First, look at the table and ask yourself, “What happened to \begin{align*}x\end{align*} (input) to get \begin{align*}y\end{align*} (output)?”

What happened to 1 to get 3? What happened to 2 to get 6? If you look carefully, you will see that the input value is multiplied by 3 to get the output value.

You can write the pattern rule as, \begin{align*}3x\end{align*}.

Next, see if the rule \begin{align*}3x\end{align*} works for each term in the table by plugging the input into the expression and seeing if it equals the listed output.

\begin{align*}\begin{array}{rcl} && 3x \\ && 3(1) \\ && 3 \ (\text{output}) \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} && 3x \\ && 3(2) \\ && 6 \ (\text{output}) \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} && 3x \\ && 3(3) \\ && 9 \ (\text{output}) \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} && 3x \\ && 3(6) \\ && 18 \ (\text{output}) \end{array}\end{align*}

#### Example 2

Write a rule for the table below.

 Input Output 2 6 3 8 4 10 5 12

First, look at the table and ask yourself, “What happened to \begin{align*}x\end{align*} to get \begin{align*}y\end{align*}?”

If you look carefully, you will see that the value in the input column was multiplied by 2 and then 2 was added to that value.

Next, let \begin{align*}x\end{align*} be the input value and write the expression.

\begin{align*}2x + 2\end{align*}

The answer is \begin{align*}2x + 2\end{align*}.

Then, see if the rule \begin{align*}2x + 2\end{align*} work for each term in the table by plugging the input into the expression and seeing if it equals the listed output.

Substitute the input values in for \begin{align*}x\end{align*} in the expression \begin{align*}2x + 2\end{align*} to see if you get the results in the output column.

\begin{align*}\begin{array}{rcl} && 2x + 2 \\ && 2(2) + 2 \\ && 4 + 2 \\ && 6 \end{array}\end{align*}
\begin{align*}\begin{array}{rcl} &&2x + 2 \\ &&2(3) + 2 \\ &&6 + 2 \\ &&8 \end{array}\end{align*}
\begin{align*}\begin{array}{rcl} &&2x + 2 \\ &&2(4) + 2 \\ &&8 + 2 \\ &&10 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} &&2x + 4 \\ &&2(4) + 4 \\ &&8 + 4 \\ &&12 \end{array}\end{align*}
The results check out.

#### Example 3

Write a rule for the table below.

 Input Output 10 6 9 5 8 4 7 3

First, look at the table and ask yourself, “What happened to \begin{align*}x\end{align*} (input) to get \begin{align*}y\end{align*} (output)?”

If you look carefully, you will see that 4 is subtracted from the input value to get the output value.

Next, let \begin{align*}x\end{align*} be the input value and write the expression.

\begin{align*}x - 4\end{align*}

The answer is \begin{align*}x - 4\end{align*}.

Then, see if the rule \begin{align*}x - 4 \end{align*} work for each term in the table by plugging the input into the expression.

Substitute the input values in for \begin{align*}x\end{align*} in the expression \begin{align*}x - 4\end{align*} to see if you get the results in the output column.

\begin{align*}\begin{array}{rcl} && x - 4 \\ && 10 - 4 \\ && 6 \ (\text{output}) \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} && x - 4 \\ && 9 - 4 \\ && 5 \ (\text{output}) \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} && x - 4 \\ && 8 - 4 \\ && 4 \ (\text{output}) \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} && x - 4 \\ && 7 - 4 \\ && 3 \ (\text{output}) \end{array}\end{align*}

#### Example 4

Write a rule for the table below.

 Input Output 2 4 4 8 6 12 7 14

First, look at the table and ask yourself, “What happened to \begin{align*}x\end{align*} (input) to get \begin{align*}y\end{align*} (output)?”

If you look carefully, you will see that the input value is multiplied by 2 to get the output value.

Next, let \begin{align*}x\end{align*} be the input value and write the expression.

\begin{align*}(x)^2\end{align*} or \begin{align*}2x\end{align*}.

The answer is \begin{align*}2x\end{align*}.

Then, see if the rule \begin{align*}2x\end{align*} works for each term in the table by plugging the input into the expression to see if you get the output value.

Substitute the input values in for \begin{align*}x\end{align*} in the expression \begin{align*}2x\end{align*} to see if you get the results in the output column.

\begin{align*}\begin{array}{rcl} && 2x \\ && 2(4) \\ && 8 \ (\text{output}) \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} && 2x \\ && 2(5) \\ && 10 \ (\text{output}) \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} && 2x \\ && 2(6) \\ && 12 \ (\text{output}) \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} && 2x \\ && 2(7) \\ && 14 \ (\text{output}) \end{array}\end{align*}

#### Example 5

Write a rule for the table below.

 Input Output 0 5 1 6 2 7 4 9

First, look at the table and ask yourself, “What happened to \begin{align*}x\end{align*} (input) to get \begin{align*}y\end{align*} (output)?”

If you look carefully, you will see that 5 is added to the input value to get the output value.

Next, let \begin{align*}x\end{align*} be the input value and write the expression.

\begin{align*}x + 5\end{align*}

The answer is \begin{align*}x + 5\end{align*}.

Now, see if the rule \begin{align*}x + 5\end{align*} work for each term in the table by plugging the input into the expression and seeing if it equals the listed output?

Substitute the input values in for \begin{align*}x\end{align*} in the expression \begin{align*}x + 5\end{align*} to see if you get the results in the output column.

\begin{align*}\begin{array}{rcl} && x + 5 \\ && 5 + 0 \\ && 5 \ (\text{output}) \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} && x + 5 \\ && 1 + 5 \\ && 6 \ (\text{output}) \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} && x + 5 \\ && 2 + 5 \\ && 7 \ (\text{output}) \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} && x + 5 \\ && 3 + 5 \\ && 8 \ (\text{output}) \end{array}\end{align*}

### Review

There are three parts to each problem below. First, write an expression for each input-output table. Next, using a variable for the value in the input column, determine the next two values in the table if the pattern were followed.

1.
 Input Output 1 4 2 5 3 6 4 7
1.
 Input Output 2 4 3 6 4 8 5 10
1.
 Input Output 1 3 2 6 4 12 5 15
1.
 Input Output 9 7 7 5 5 3 3 1
1.
 Input Output 8 12 9 13 11 15 20 24
1.
 Input Output 3 21 4 28 6 42 8 56
1.
 Input Output 2 5 3 7 4 9 5 11
1.
 Input Output 4 7 5 9 6 11 8 15
1.
 Input Output 5 14 6 17 7 20 8 23
1.
 Input Output 4 16 5 20 6 24 8 32

To see the Review answers, open this PDF file and look for section 12.10.

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Color Highlighted Text Notes

### Vocabulary Language: English

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.

Input-Output Table

An input-output table is a table that shows how a value changes according to a rule.

Pattern

A pattern is a series of pictures, numbers or other symbols that repeat in some way according to a rule.