Mrs. Hawk’s sixth grade class was so motivated by the idea of taking a coach bus on their class trip that they had 100% turn out for the car wash on Saturday. The students gathered their supplies and washed cars for most of the day. The car wash started at 9 am and continued until 2 pm.
The students figured out that they needed to earn $244.92. To make the math easier, they rounded up to $245.00. At $5.00 a car, they needed to wash 49 cars to make enough money for the bus.
The car wash was a busy place. At the beginning there weren’t any cars, but between 9 am and 10 am the class washed 5 cars. From 10 to 11, the class washed 10 cars, from 11 to 12 the class washed 15 cars and from 12 – 1 the class washed 20 cars.
Toby kept track of all of this information in his notebook. He created a chart to show how the number of cars washed changed throughout the day.
\begin{align*}&0 \qquad 0\\ &1 \qquad 5\\ &2 \qquad 10\\ &3 \qquad 15\\ &4 \qquad 20\end{align*}
Toby can see a pattern in the data, can you? In this Concept you will learn how to write rules for patterns. Pay close attention and at the end of this Concept you will have chance to write a rule, an expression that matches this table.
Guidance
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 made or created or counted and then they can draw conclusions based on those patterns.
A pattern functions according to a rule. In this Concept, we are going to be looking at different patterns and at how to decipher and write rules for patterns.
What is a pattern?
A pattern is something that repeats in a specific way. A pattern functions according to a rule. The rule tells us how the pattern repeats.
We can look at patterns in nature-for example the number of leaves on a flower or the number of branches on a tree are special patterns.
2, 4, 6, 8, 10.....
Once you have a pattern, we can establish a rule about the pattern. This pattern counts by two’s. We could say that we add two to each previous term to get the next term in the pattern.
How can we write this so that anyone could understand the rule?
In this example, we could use a variable to represent the terms in the list. Let’s use \begin{align*}x\end{align*}.
\begin{align*}x=\end{align*} term in the pattern
By term we mean the numbers 2, 4, 6 and so on.
Next, we can add more to the variable. Since we add two to each term to get the next term, then we can say that \begin{align*}x\end{align*} plus two is the rule.
Rule: \begin{align*}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 I take 2 and substitute it for \begin{align*}x\end{align*} then 2 + 2 = 4, so the rule works.
If I take 4 and substitute it for \begin{align*}x\end{align*} then 4 + 2 = 6, so the rule works.
If I take 6 and substitute it for \begin{align*}x\end{align*} then 6 + 2 = 8, so the rule works.
Is there an easier way to figure this out?
Yes. We can use a table. We call it an input/output table.
Input | Output |
---|---|
2 | 4 |
4 | 6 |
6 | 8 |
8 | 10 |
Let’s see if our 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.
Does the rule \begin{align*}x+2\end{align*} work for each term in the table?
Yes it does. Two can be added to each term in the input column to equal the output column.
You can write rules by examining the patterns in input/output tables.
Input | Output |
---|---|
0 | 0 |
1 | 3 |
2 | 6 |
3 | 9 |
What happened to the input to get the output?
This is where we can look at figuring out a rule. It is a little like deciphering a puzzle. You have to think of what happened to one term to equal another term.
The term in the input column was multiplied by 3 to get the number in the output column. This is the rule for this table.
We 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.
Rule \begin{align*}= 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 |
What is the rule of this table? What happened to the input to get the output?
This is tricky, but if you look for patterns you will see that the input was multiplied by two and then one was added. We can write the rule as an expression.
If you think of the input as a variable, we can write a rule for this table that looks like this.
Rule \begin{align*}= 2x+1\end{align*}
We call the input-output relationship of terms a function.
You will learn all about functions in the next Concept.
Now it's time for you to practice. Write an expression for each input - output table.
Example A
Input | Output |
---|---|
10 | 6 |
9 | 5 |
8 | 4 |
7 | 3 |
Solution: \begin{align*}x - 4\end{align*}
Example B
Input | Output |
---|---|
2 | 4 |
4 | 8 |
6 | 12 |
7 | 14 |
Solution: \begin{align*}2x\end{align*}
Example C
Input | Output |
---|---|
0 | 5 |
1 | 6 |
2 | 7 |
4 | 9 |
Solution: \begin{align*}x + 5\end{align*}
Now back to the dilemma of the car wash.
Here is the original problem once again. Use what you have learned to write a rule for the pattern of cars washed.
Mrs. Hawk’s sixth grade class was so motivated by the idea of taking a coach bus on their class trip that they had 100% turn out for the car wash on Saturday. The students gathered their supplies and washed cars for most of the day. The car wash started at 9 am and continued until 2 pm.
The students figured out that they needed to earn $244.92. To make the math easier, they rounded up to $245.00. At $5.00 a car, they needed to wash 49 cars to make enough money for the bus.
The car wash was a busy place. At the beginning there weren’t any cars, but between 9 am and 10 am the class washed 5 cars. From 10 to 11, the class washed 10 cars, from 11 to 12 the class washed 15 cars and from 12 – 1 the class washed 20 cars.
Toby kept track of all of this information in his notebook. He created a chart to show how the number of cars washed changed throughout the day.
\begin{align*}&0 \qquad 0\\ &1 \qquad 5\\ &2 \qquad 10\\ &3 \qquad 15\\ &4 \qquad 20\end{align*}
Toby can see a pattern in the data, can you?
Each number in the left hand column shows the time that passed.
In the beginning there weren’t any cars.
Then in the first hour the students washed 5 cars.
In the second hour, they washed 10 cars.
In the third hour, they washed 15 cars.
In the fourth hour, they washed 20 cars.
If we wanted to write a rule for the pattern, what happened to the input to get the output?
The input was multiplied by 5.
The rule for the number of cars washed per hour is \begin{align*}5x\end{align*}.
Given this rule, how many cars can we predict will be washed in the fifth hour?
Write down your prediction and check it with a friend.
If each car paid $5.00, how much money did the students make in five hours?
75 \begin{align*}\times\end{align*} 5 \begin{align*}=\end{align*} $375.00
The students are very excited! They will be able to take the coach bus to the amusement park!!
Vocabulary
Here are the vocabulary words in this Concept.
- Pattern
- a series of pictures, numbers or other symbols that repeats in some way according to rule.
- Function
- one variable depends on the other and there is only one output for each input in a function.
- Input-Output Table
- A table that shows how a value changes according to a rule.
Guided Practice
Here is one for you to try on your own.
Write a rule for the table.
Input | Output |
---|---|
2 | 6 |
3 | 8 |
4 | 10 |
5 | 12 |
Answer
In looking at this table, we can ask ourselves, "What happened to x to get y?"
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.
This is our rule. We can write it as an expression.
\begin{align*}2x + 2\end{align*}
This is our answer.
Video Review
Here are videos for review.
James Sousa, Introduction to Functions, Part 2
James Sousa, Example of Writing a Cost Function and Completing a Table of Values
Practice
Directions: Write an expression for each input-output table. Use a variable for the value in the input column of the table.
Then add the next two values in the table if the pattern were followed. There are three parts to each answer.
1.
Input | Output |
---|---|
1 | 4 |
2 | 5 |
3 | 6 |
4 | 7 |
2.
Input | Output |
---|---|
2 | 4 |
3 | 6 |
4 | 8 |
5 | 10 |
3.
Input | Output |
---|---|
1 | 3 |
2 | 6 |
4 | 12 |
5 | 15 |
4.
Input | Output |
---|---|
9 | 7 |
7 | 5 |
5 | 3 |
3 | 1 |
5.
Input | Output |
---|---|
8 | 12 |
9 | 13 |
11 | 15 |
20 | 24 |
6.
Input | Output |
---|---|
3 | 21 |
4 | 28 |
6 | 42 |
8 | 56 |
7.
Input | Output |
---|---|
2 | 5 |
3 | 7 |
4 | 9 |
5 | 11 |
8.
Input | Output |
---|---|
4 | 7 |
5 | 9 |
6 | 11 |
8 | 15 |
9.
Input | Output |
---|---|
5 | 14 |
6 | 17 |
7 | 20 |
8 | 23 |
10.
Input | Output |
---|---|
4 | 16 |
5 | 20 |
6 | 24 |
8 | 32 |