# Translating Between Tables & Expressions

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

We can make an expression (or rule) from a table of ordered pairs. Given a sequence of numbers as ordered pairs, we can find an expression to represent this sequence. Furthermore, we can use such expressions to represent numerical relationships in real-world problems.

## Creating an Expression to Represent a Table of Ordered Pairs

Let's say that we have 2 runners. Let's say that they run at the same pace or speed. If the 1st runner gets a 1-mile head start on the other, can we create an expression or rule to represent their positions in relation to each other through any number of miles?

### A Sequence of Ordered Pairs

Before we can find an expression to represent the 2 runners' positions in relation to one another, we should define what an ordered pair is. An ordered pair is essentially two numbers that are paired together in an ordered sequence with other pairs of numbers. Note that we call it a sequence, because these pairs follow a set pattern or rule. For example, we might have a sequence of ordered pairs as follows:

In this table, the 1st column gives us our position of a n term in a sequence of numbers. The 2nd column represents a value in another sequence of numbers, dependent on the numbers in the 1st column. Each row of two numbers in this table is an ordered pair. Looking at the sequence, we see that for every increase of 1 in the 1st column, there is an increase of 4 in the 2nd column.

#### Developing an Expression

We can develop an expression to represent the relationship between these rows or ordered pairs. Again, for every increase of 1 in the 1st position column, there is an increase of 4 in the 2nd value column, which means that whatever the number is in the 1st column, it gets multiplied by 4; therefore, we can say that whatever number n represents, it will be multiplied by 4.

For example, in the 5th row, we can say that 4n equals 20 or 4n = 20. Note our expression is 4n. To find out what n equals in the 5th row down, we can divide both sides of 4n = 20 by 4: 4n / 4 = 20 / 4, which is 1n = 5 or n = 5. In other words, the value of n in the 1st column of the 5th row is 5. If we wished to check to see if our expression 4n is true for any position, we can place any number in the 1st column into n and we will get its match for its ordered pair on the right.

For instance, when n = 1, 4(1) = 4. When n = 2, 4(2) = 8, etc. In other words, in row 1, we have an ordered pair of 1 and 4. In row 2, we have an ordered pair of 2 and 8. Now we know that our expression is true!

### Ordered Pairs in a Real-World Relationship

These pairs of numbers in the last table might represent some real-world relationship -now refer to the table below. For instance, the numbers in the 1st column might represent some number of cars, while the numbers in the 2nd column might represent the total number of wheels on those total number of cars. Remember that each row in the table below represents an ordered pair or a relationship.

1 car has 4 wheels. 2 cars have 8 wheels and so on. Using the expression 4n that we found earlier, we have a rule to represent the total number of tires that we need for a total n number of cars. We can say that for every n cars, there are 4 wheels; therefore, if we have n = 5 cars and 4n = 20, then 4 wheels (5 cars) = 20 wheels. In other words, 5 cars have 20 wheels. Note that this expression 4n will hold true for any number we put in for n. So, for any number of cars, we can determine a total amount of tires!

### Making Expressions for Problems

Since we have learned what an expression is and how it may be used for a real-world problem, we can look back at our very first problem with the runners:

#### Two Runners' Positions in Relation to One Another

Say that we have 2 runners. Suppose that they run at the same pace or speed. Suppose that the 1st gets a mile head start over the other. We would have the following sequence of ordered pairs to represent where the runners are with respect to one another:

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